derivativewise unramified infinite integral extensions

JOURNAL OF ALGEBRA 136, 197-247 (1991)

Derivativewise Unramified infinite Integral Extensions

SHREERAM S. ABHYANKAR* ANI) WILLIAM J. HEINZER~

Department of Mathematics, Purdue University, West Lafayette, Indiana 47907

Communicated by Nagayoshi Iwahori

Received January 15, 1989

Let A be a normal noetherian domain with quotient field K and let B be a localization of the integral closure of A in an infinite algebraic field extension of K. Two obvious necessary conditions in order that B be noetherian are finite splitting and finite ramification of prime ideals of A in B. We consider various situations in which these conditions are also sufficient. As an important ingredient of this we give conditions for a prime ideal in B to be the extension of its contraction in A, When B is noetherian we investigate preservation from A to B of the property of being pseudogeometric. We do all this by introducing the concept of compositumwise unramifiedness and various related notions. We also present structure theorems for compositumwise unramified extensions in the local case. This local theory ensures finite splitting of all prime ideals by assuming it only for maximal ideals. 0 19% Academic Press, Inc.

1. INTRODUCTION

Let A be a normal noetherian domain with quotient field K and let B be the integral closure of A in an algebraic field extension L of K which is not necessarily finite. We are interested in the following general questions.

General Questions. (1) Under what conditions does it follow that B is noetherian? Locally noetherian?

(2) When does A pseudogeometric imply 3 is pseudogeometric? (3) When does A regular imply B is regular?

* Abhyankar’s work was partly supported by NSF Grant DMSSg-16286, ONR Grant NOOO14-88-K-0689, ONR Grant N00014-86-0689 under URI, and ARO Contract DAAG29-85-C-0018 under Cornell MSI, at Purdue.

+ Heinzer’s work was partly supported by NSF Grant DMS-8800762, at Purdue

197 0021-8693/91 $3.00

Copyright 0 1991 by Academic Press, Inc. All rights of reproduction in any fmm reserved.

198 ABHYANKAR AND HEINZER

(4) When does the singular locus of A closed imply that the singular locus of B is closed?

(5) When does A locally analytically normal (or analytically irreducible) imply B is locally analytically normal (or analytically irreducible)?

(6) When does A excellent imply B is excellent? (7) When does A a UFD imply B is a UFD?

The present paper deals with questions (1) and (2). In forthcoming papers we plan to discuss the remaining questions. The three forthcoming papers in this sequence being:

Second paper: Singular locus of an infinite integral extension, Third paper: Ramification in infinite integral extensions, Fourth paper: Examples and counterexamples in commutative ring

theory.

The entire series of these four papers deals with the theme of Integral Closure and Ramification of Prime Ideals in Infinite Algebraic Field Extensions.

In considering the above questions we find it expedient to relax the hypothesis on A and B to the case when A and B need not be domains and B need not be integral over A. Instead we assume that B is an overring of an arbitrary ring A (always commutative with 1) such that B is “almost compositumwise unramified” over A at various primes of B in the following sense.

Given any prime ideal Q in B, an element x of B is said to be derivativewise unramified over A at Q if there exists a manic polynomial f(X) with coefficients in A such that f(x) = 0 but f’(x) I$ Q; moreover, B is said to be compositumwise unramified over A at Q if there exists a subset W of B such that every element of W is derivativewise unramified over A at Q, and B is the localization of A[ W] at some multiplicative set in A[ W]. Certain weaker versions of this are called almost compositumwise unramifiedness and weakly almost compositumwise unramifiedness. When these properties hold “uniformly” on a set of primes 2 of B, we respectively say that B is uniformly almost compositumwise unramihed over A at 2, or B is uniformly weakly almost compositumwise unramified over A at 2. The details of these and other definitions are given in Section 2.

In part (6.14.1) of Theorem (6.14) of Section 6 we prove that if A is a normal noetherian domain and B is a domain which is almost compositumwise unramified over A at each prime ideal of B and for every nonzero prime ideal P of A we have that B is uniformly almost compositumwise unramified over A at the set of all prime ideals of B containing P, then: B is noetherian if and only if every prime ideal in A is finitely split

UNRAMIFIED INTEGRAL EXTENSIONS 199

in B, i.e., if and only if at most a finite number of prime ideals in B contract to any given prime ideal in A.

In (6.14.3) we prove that if B is normal, then the uniformity hypothesis in (6.14.1) need only be assumed for prime ideals in A whose height is greater than one. In (6.145) we show that if B is normal and the dimension of A is at most 2, then the uniformity hypothesis of (6.14.1) can be completely dispensed with.

In (6.14.2), (6.14.4), and (6.14.6) we observe that if B is integral over A, then in the finite splitting condition above we may restrict our attention to maximal ideals. This follows because our results in Section 4 imply that B is locally noetherian at each maximal ideal under the hypothesis of (6.14).

In (6.14) we also show that B is noetherian if instead of assuming A to be normal we assume that A is pseudogeometric and the hypothesis of (uniformly) almost compositumwise unramifiedness is replaced by the hypothesis of (uniformly) weakly almost compositumwise unramifiedness.

In relation to question (2), we consider in (6.15) and (6.15A) the preservation of the pseudogeometric property from A to B. Likewise, in (6.13) we consider the preservation of normality from A to B.

In Theorems (4.13), (4.14), and (4.15) of Section 4 we establish certain local versions of Theorems (6.13), (6.14), and (6.15), respectively. In Propositions (4.1) and (4.5) to (4.8) of Section 4 we give structure theorems for compositumwise unramilied extensions in the local case.

In Section 4 we use Cohen’s theory of generalized local rings (glrs) which we review in Section 3. On the other hand the proofs in Section 6 are based on Section 5 where we give conditions for a prime ideal in B to be the extension of its contraction in A. In the typical situation considered in Section 5, B is assumed to be a localization of the homomorphic image of the univariate polynomial ring (i.e., of the polynomial ring in one variable) A[X] over A modulo a manic polynomial. In Section 6, we also use the noetherian theorems of Cohen and Eakin which we review in Section 3.

In the forthcoming third paper of this series, entitled “Ramification in Infinite Integral Extensions,” we shall strengthen the above structure theorem (4.7) by proving that if R is a normal local domain and S is a normal quasilocal domain lying above R such that S is separable algebraic and unramified over R, then S is compositumwise unramified over R. Ia fact we shall prove this under the weaker hypothesis that R is a normal sublocal domain, where by a sublocal domain we mean a quasilocal domain which is dominated by a local (noetherian) domain..

In the second paper, entitled “Singular Locus of an Infinite Integral Extension,” we let B be the integral closure of the bivariate polynomial ring A = F[X, Y] in a suitable infinite algebraic field extension of F(X, Y) where F is the field of rational numbers. We obtain in this manner a normal noetherian domain B of dimension 2 such that B is pseudogeometric

200 ABHYANKARAND HEINZER

and locally excellent, but the singular locus of B consists of an infinite number of maximal ideals without containing any nonmaximal prime ideal and hence is not closed in the Zariski topology. So in particular, B cannot be excellent.

In the fourth paper we shall construct numerous examples which examine the limitations of the hypotheses made in various situations in the first three papers. For instance in connection with Remark (5.3) of Section 5, we construct an example of a principal ideal I in the univariate polynomial ring A[X] over a domain A such that I is generated by a manic polynomial and such that I has a primary decomposition which expresses Z as an intersection of two prime ideals neither of which is finitely generated. As another illustration of limitations of hypothesis, in connection with Proposition (4.7) we point out that a normal local domain S lying above a one-dimensional local domain R need not be compositumwise unramified over R even when it is unramihed and integral over R.

2. TERMINOLOGY

(2.1) Polynomials. Throughout this paper, X denotes an indeterminate. The X-derivative of a polynomial f(X) is denoted by f’(x). A polynomial f(X) is said to be manic iff(X) is nonzero and in it the coefficient of the highest degree term is 1. A polynomialf(X) is said to be nonconstant if (it is nonzero and) its degree is positive.

(2.2) Cofinal families. By a cofinal family of subsets of a set S we mean an indexed family (R,),, E of subsets of S such that every finite subset of S is contained in R, for some e E E. A family of subsets (RJeeE of a set S is said to be cojkal in S if it is a colinal family of subsets of S. We may also express this by saying that the set S is a cofinal union of the sets U’LLE.

(2.3) Rings. By a ring we mean a commutative ring with 1. For a ring A, by dim A we denote the (Krull) dimension of A; note that then dim A is either a nonnegative integer or cc or - co; moreover, dim A = - CO o A is the null ring; on the other hand, dim A = co *for every nonnegative integer d there exists a chain of prime ideals

PocP,c *.. CPd inAwithP,#P,# ... #Pd.

A ring A is said to be normal if it is integrally closed in its total quotient ring. A domain ( = integral domain) B is said to be separable algebraic over a subring A if the quotient field of B is a separable algebraic extension of the quotient field of A.


Given a subring A of a ring B, by an A-module basis of B we mean a family (x,),,~ of elements of B such that every x E B can be expressed as a sum x=CeEE a,x, with a, E A (and a, = 0 for all except a finite number of e); such a family is said to be a free A-module basis of B if, moreover, c esE a,x, = 0 with a, E A implies a, = 0 for all e. Given a subring A of a ring B, B is said to be afinite (resp. free, finite-free) A-module if there exists a finite (resp. free, finite and free) A-module basis of B.

A ring A is said to be prequasipseudogeometric if A is a domain and the integral closure of A in any finite algebraic field extension of the quotient field of A is a finite A-module. A ring A is said to be quasipseudogeometric if for every prime ideal P in A we have that A/P is prequasipseudogeometric. A ring A is said to be prepseudogeometric if A is noetherian and prequasipseudogeometric. A ring A is said to be pseudogeometric if A is noetherian and quasipseudogeometric. We take this opportunity to point out that the first line of the third paragraph on p. 11 of Abhyankar [3-j which now reads “A ring A is said to be pseudogeometric if for every prime ideal’” should be changed to read “A ring A is said to be pseudogeometric if A is noetherian and for every prime ideal.”

A ring A is said to be ideally (resp. prime ideally, finitely generated ideally, principal ideally, nonzerodivisor principal ideally) closed in an overring B if for every (resp. every prime, every finitely generated, every principal, every nonzerodivisor principal) ideal I in A we have (IB) n A = I; note that by a nonzerodivisor principal ideal in A we mean an ideal of the form aA where a is a nonzerodivisor in A.

(2.4) Zerosets. Given any ring A, for any IE A or ic A, we define Z(I, A) - the zeroset (or, the variety) of I in A = the set of all prime ideals P in A such that IA c P, and we define MZ(I, A) = the maximal zeroset of I in A = the set of all maximal ideals P in A such that IA c P, and we note that members of MZ(I, A) are exactly the maximal, members of Z(I, A), and we define mZ(I, A) = the minimal zeroset of I in A = the set of all minimal members of Z(I, A); i.e., mZ(1, A) = the set of all P E Z(I, A) such that for every P’ E Z(I, A) with P’ c P we have P’= P, and we note that members of mZ(I, A) are called the minimal prime idedls of I in A, and we define N(I, A) = the strong complement of Z in A = nPtZC,,Aj (A\P), and we note that if IA #A then N(I, A) is a multiplicative set in J1. Recall that for a multiplicative set N in a ring A, it is required that 0 6 N and 1 EN.

Given any subring A of a ring B, for any PC A, and for any JE B CC JC B, we define Z(J, B; P, A) = the zeroset of J in B lying over P in A={QEZ(J,B):Q~A=P}.

Given any subring A of a ring B, a prime ideal Q in B is said to be unsplit (resp.j%itely split) over A if upon letting P = Q n A we have Z(0, B; P, A) = (Q} (resp. Z(0, B; P, A) = a finite set).

202 ARHYANKAR AND HEINZER

Given any subring A of a ring B, a prime ideal P in A is said to be unsplit (resp. finitely split) in B if Z(0, B; P, A) contains at most one element (resp. Z(0, B; P, A) = a finite set).

(2.5) Quasilocal rings. By a quasilocal ring we mean a ring having exactly one maximal ideal. The maximal ideal in a quasilocal ring R is denoted by M(R). By a local ring we mean a noetherian quasilocal ring. Given quasilocal rings R and S, we say that S dominates R if R is a subring of S and M(R) c M(S).

Given a quasilocal ring S and a subring D of S, upon letting v : S + S/M(S) be the canonical epimorphism, we say that S is residually rational over D if v(S) = the quotient field of v(D), and we say that S is residually separable algebraic over D if v(S) is separable algebraic over Q )+

An ideal I in a quasilocal ring R is said to be closed (in R) if n 2 o (I+ M(R)‘) = I. Following Cohen, by a glr ( = generalized local ring) we mean a quasilocal ring in which the zero ideal is closed and the maximal ideal is finitely generated. Note that by Krull’s intersection theorem, a local ring is a glr. The completion of a glr R is denoted by i?; we may regard ri to be an overring of R; we note that then ff dominates R, I? is residually rational over R, and M(R) l? = M(R). R is said to be analytically irreducible if & is a domain; R is said to be analytically normal if fi is a normal domain.

(2.6) Localizations. Let N be a multiplicative set in a ring A. For any IE A or IC A, we define j*[I, N, A] = the isolated component of

I at N in A={aEA:naEIA for some nEN}, and we note that IA cj*[J N, A] = an ideal in A, and we define Z*(I, N, A) = the zeroset ofIatNinA={QEZ(I,A}:NnQ=@},andMZ*(I,N,A)= themaxi- ma1 zeroset of1 at N in A = the set of all maximal members of Z*(I, N, A), i.e., MZ*(I, N, A) = the set of all P&Z*(I, N, A) such that for every P’ E Z*(I, N, A) with P c P’ we have P = P’, and we define mZ*(I, N, A) = the minimal zeroset of I at N in A= (QEmZ(I, A): NnQ=@} and we note that members of mZ*(I, N, A) are exactly the minimal members of Z*(I, N, A), i.e., mZ*(I, N, A) = the set of all P E Z*(I, N, A) such that for every P’ E Z*(I, N, A) with P’ c P we have P’ = P; for any prime ideal P in A we define j*[I, P, A] = the isolated component of I at P in A = j*[I, A\P, A], Z*(I, P, A)= the zeroset of I at P in A=Z*(I, A\P, A), MZ*(I, P, A) = the maximal zeroset of I at P in A = MZ*(I, A\P, A), and mZ*(I, P, A) = the minimal zeroset of I at P in A = mZ*(I, A\P, A).

If every element of N is invertible in an overring B of A, then by the localization of A at N in B we mean the subring A,,, of B obtained by putting A, = (b E B: nb E A for some n E N} note that the existence of such an overring B implies that N does not contain any zerodivisor of A; when


the reference to B is clear from the context, it need not be made explicit. If N does not contain any zerodivisor of A, then we can form A, in the total quotient ring of A. If P is a prime ideal in A such that every element of the multiplicative set A\P is invertible in the overring B of A, then by the localization of A at P in B we mean the localization of A at A\P in B and we may denote it by A, instead of A,,,; again when the reference to B is clear from the context, it need not be made explicit.

In the general case (i.e., when N is allowed to contain zerodivisors of A), the image of N under the canonical epimorphism A -+ A/‘j*[O, N, A] is a multiplicative set in A/j*[O, N, A] and every element of the said image is invertible in the total quotient ring of A/j*[O, IV, A], and so we may form the localization A, of A/‘j*[O, N, A] at the said image in the said total quotient ring; this ring A, is called the canonical localization of A at N and we let jCA, N] : A -+ A, be the composition of the maps A + A/j*[O, N, A] -+ A, where the first map is the canonical epimorphism and the second map is the canonical monomorphism; we call j[A, N] the canonical localization map of A at N, by a localization map of A at N we mean a ring homomorphism j : A --f A’ such that K.er j =j* [O, N, A] and A’ is the localization of j(A) at j(N) in A’; we note that then there exists a unique A-isomorphism A’ -+ A,. Again, in the general case, for any prime ideal P in A, by the canonical localization of A at P and the ~a~o~~~a~ localizatiotz map of A at P we mean the canonical localization of A at A\P and the canonical localization map of A at A\P and we may denote these by AP and j[A, P] instead of AA,P and j[A, A\P], respectively; finally, by a localization map of A at P we mean a localization map of A at

(2.7) Essential properties. A ring C is said to be a localization of a subring B if there exists a multiplicative set N in B such that (every element of N is invertible in C and) C is the localization of B at N in 6; note that if C is quasilocal then: C is a localization of the subring B -=s- C is the localization of B at M(C) n B in C.

A ring C is said to be essentially integral (resp. essentially finite, essel;o- tially free, essentially finite-free) over a subring A if there exists a subring B of C with A c B such that C is a localization of B and such that B is integral over A (resp. B is a finite A-module, B is a free A-module, B is a finite-free A-module).

A domain C is said to be almost finite over a subring A if C is essentially integral over A and the quotient field of C is a finite algebraic field extension of the quotient field of A.

Given quasilocal domains A and C, we say that C lies above A if C dominates A and C is essentially integral over A.

Given a multiplicative set N in a ring A, we say that A is normal (msg. noethtrian, prequasipseudogeometric, quasipseudogeometric, prepseudo-


geometric, pseudogeometric) at N if some (and hence every) localization of A at N is normal (resp. noetherian, prequasipseudogeometric, quasipseudogeometric, prepseudogeometric, pseudogeometric). Given a prime ideal P in a ring A, we say that A is normal (resp. noetherian, prequasipseudogeometric, quasipseudogeometric, prepseudogeometric, pseudogeometric) at P if A is normal (resp. noetherian, prequasipseudogeometric, quasipseudogeometric, prepseudogeometric, pseudogeometric) at A\P. Given a ring A and Z c Z(0, A), we say that A is normal (resp. noetherian, prequasipseudogeometric, quasipseudogeometric, prepseudogeometric, pseudogeometric) at 2 if for every P E 2 we have that A is normal (resp. noetherian, prequasipseudogeometric, quasipseudogeometric, prepseudogeometric, pseudogeometric) at P.

(2.8) Unramlj?edness. Let A be a subring of a ring B. Given XE B we say that x is derivativewise unram$ed for B over A if

there exists a manic polynomial f(X) E A[X] such that f(x) = 0 and f(x) is invertible in B. Given x E B and a prime ideal Q in B, we say that x is derivativewise unramified for B over A at Q if there exists a manic polynomial f(X) E A[X] such that f(x) = 0 and f’(x) $ Q.

Given WC B we say that W is derivativewise unramiJed for B over A if every x E W is derivativewise unramified for B over A. Given W c B and a prime ideal Q in B, we say W is derivativewise unramified over A at Q if every x E W is derivativewise unramified over A at Q.

We say that B is elementwise unramgied over A if there exists x E B such that x is derivativewise unramified for B over A and B is a localization of A[x]. Given a prime ideal Q in B we say that B is elementwise unramified over A at Q if there exists XE B such that x is derivativewise unramified over A at Q and B is a localization of A[x]. Given 2 c Z(0, B) we say that B is elementwise unram$ed over A at Z if for every Q E Z we have that B is elementwise unramified over A at Q.

We say that B is compositumwise unramfied over A if there exists WC B such that W is derivativewise unramified for B over A and B is a localization of A[ W]. Given a prime ideal Q in B we say that B is compositumwise unramiJied over A at Q if there exists WC B such that W is derivativewise unramified over A at Q and B is a localization of A[ W]. Given Z c Z(0, B) we say that B is compositumwise unramified over A at Z if for every Q E Z we have that B is compositumwise unramified over A at Q.

We say that B is finite-compositumwise unramified over A if there exists a finite set W c B such that W is derivativewise unramitied for B over A and B is a localization of A[ W]. Given a prime ideal Q in B we say that B is finite-compsitumwise unramljied over A at Q if there exists a finite set WC B such that W is derivativewise unramified over A at Q and B is a localization of A[WJ Given ZC Z(0, B) we say that B is finite-

UNRAMIFIJSD INTEGRAL EXTENSIONS 205

compositumwise unramified over A at Z if for every QEZ we have that B is finite-compositumwise unramified over A at Q.

Given any Q EZ(O, B) we say that B is almost composidumwise anramified over A at Q if there exists a subring A’ of B with A c A’ such that A’ is finite-compositumwise unramified over A at mZ(0, A’), and B is compositumwise unramified over A’ at Q. Given any 2 c Z(0, B) we say that B is almost compositumwise unramified over A at Z if for every Q E Z we have that B is almost compositumwise unramified over A at Q. Given any ZC Z(0, B) we say that B is uniformly almost compositumwise unramified over A at Z if there exists a subring A’ of B with A c A’ such that A’ is finite-compositumwise unramilied over A at mZ(0, A’), and B is compositumwise unramified over A’ at 2.

Given any Q E Z(0, B) we say that B is weakly almost compositumwise unramified over A at Q if there exists a subring A’ of B with A c A’ such that A’ is essentially finite over A, and B is compositumwise unramified over A’ at Q, Given any Z c Z(0, B) we say that B is weakly almost compositumwise unramified over A at Z if for every QEZ we have that B is weakly almost compositumwise unramified over A at Q. Given any Zc Z(0, B) we say that B is uniformly weakly almost compositumwise unramified ouer A at Z if there exists a subring A’ of B with A c A’ such that A’ is essentially finite over A, and B is compositumwise unramified over A’ at Z.

In case B is quasilocal, we say that B is unramified over A if (M(B) A A) B = M(B) and B is residually separable algebraic over A.

In the general case, given any Q E Z(0, B), we say that B is unram@ed over A at Q if, upon letting j: B -+ B, be a localization map of B at Q, we have that B, is unramified over j(A). Again in the general case, given any Z c Z(0, B), we say that B is unramified over A at Z if B is unramified over A at every Q E Z.

Concerning the definitions made in (2.8), the following two Observations (2.9) and (2.10) may be used tacitly.

Observation (2.9) Given any subring A’ of a ring B and given any subring A of A’, we obviously have the following: If XE B is derivativewise unramified for B over A, then clearly x is derivativewise unramified for B over A’. If W c B is derivativewise unramified for B over A, then W is derivativewise unramified for B over A’. If B is elementwise (resp. compositumwise, finite-compositumwise) unramified over A, then B is elementwise (resp. compositumwise, finite-compositumwise) unramified over A’. If B is quasilocal and B is unramified over A then B is unramified over A’. Finally, if B and A’ are quasilocal, B dominates A’, B is unramitied over A’, and A’ is unramified over A, then B is unramified over A.

Observation (2.10) Given any subring D of a quasilocal ring S, upon


letting P = M(S) n D and u : S -+ SIPS be the canonical epimorphism, we note that P is a prime ideal in D with (PS) AD = P, and, upon letting R be the localization of D at P in S, we note that R is a quasilocal subring of S, M(S) n R = M(R) = (PS) A R, M(R) S = PS, u(R) is the quotient field of u(D), and moreover, S is unramified over D * S is unramified over R o u(S) is a separable algebraic field extension of u(R) o u(S) is unramified over u(R) o u(S) is unramified over u(D). We also note that if v : S -+ S’ is any ring epimorphism with Ker v c (M(S) n D) S then S is unramified over Do S’ is unramified over v(D).

Some more observations concerning (2.8) are given in the following lemma.

LEMMA (2.11) Let A be a subring of a domain B, and let Zc Z(0, B). Then we have the following.

(2.11.1) Given any subrings C and A’ of B with A c C c A’ such that A’ is a localization of C, we have that B is compositumwise unramtj?ed over A’ at Z tff B is compositumwise unramified over C at Z.

(2.11.2) B is uniformly weakly almost compositumwise unramified over A at Z iff there exists a subring C of B with A c C such that B is compositumwise unramified over C at Z, and C is a finite A-module.

(2.11.3) B is uniformly almost compositumwise unramified over A at Z tff there exists a subring C of B with A c C such that B is compositumwise unramified over C at Z, C is a finite A-module, an C is separable algebraic over A.

(2.11.4) If Z= Z1 v . ‘. v Z, where Z,, . . . . Z, are a finite number of subsets of Z(0, B) such that for i = 1, . . . . r we have that B is untformly weakly almost compositumwtse unramtjied over A at Zi, then B is untformly weakly almost compositumwise unramtfied over A at Z.

(2.11.5) If Z= Z1 v ‘. . v Z, where Z,, . . . . Z, are a Jinite number of subsets of Z(0, B) such that for i = 1, . . . . r we have that B is uniformly almost compositumwise unramified over A at Zi, then B is untformly almost compositumwise unramijiied over A at Z.

Proof The “if” part of (2.11.1) follows from (2.9). To prove the “only if” part of (2.11.1), assume that B is compositumwise unramified over A’ at Z. Now given any Q E Z, there exists WC B and a multiplicative set N in B such that W is derivativewise unramified over A’ at Q, and B is the localization of A’[W] at N (in the quotient held of B). Also there exists a multiplicative set N’ in A’ such that A’ is the localization of C at N’. Given any x E W, there exists a manic polynomial f(X) of positive degree d in X with coefficients in A’ such that f(x) =0 and f’(x)+ Q. Since A’ is the

UNRAMIFIED INTEGRAL EXTENSIONS

localization of C at iv’, we can find be N’ such that bf(X)~ C[X]. Let g(X) = @f(X/b). Now g(X) is a manic polynomial of degree d in X with coefficients in C, and we have g(bx) =bdf(x) =O. Also g’(btx) = bd- ‘j”(x) $ Q because b is invertible in B and hence b # Q. Thus, upon letting V(X) = b, we have that v(x) EN’ and v(x) x is derivativewise unramified over C at (2. Let W* = {y(x) x : x E W} and N* = NN’ = the smallest multiplicative set in B containing N and N’. Then W* is derivativewise unramified over C at Q, and B is the localization of CC W*] at N*. Therefore B is compositumwise unramified over C at Q. This being so for every Q E 2, we conclude that B is compositumwise unramified over C at Z.

The “if” part of (2.11.2) is obvious by taking A’ = C in the relevant definition in (2.8). To prove the “only if” part of (2.11.2), assume that is uniformly weakly almost compositumwise unramified over A at Z. Now by definition there exists a subring C of B together with a multiplicative set N’ in C such that C is a finite A-module and p3 is compositumwise unramified over A’ = C,, at Z. By (2.11.1) it follows that B is compositumwise unramilied over C at Z.

Again, the “if” part of (2.11.3) is obvious by taking A’= C in t relevant definition in (2.8). To prove the “only if” part of (2.11.3), assume that B is uniformly almost compositumwise unramified over A at Z. Now by definition there exists a subring C of B together with a multiplicative set N’ in C such that C is a finite A-module, C is separable algebraic over A, and B is compositumwise unramified over A’ = CNt at Z. follows that B is compositumwise unramified over C at Z.

In view of (2.11.2) , under the hypothesis of (2.11.4), for i = 1, . . . . Y, there exists a subring C, of B with A c Ci such that B is compositumwise unramified over Ci at Zi, and Cj is a finite A-module. Upon letting C= AIC1, ..~, C,], in view of (2.9) we see that B is compositumwise unramified over C at Z, and obviously C is a finite A-module. Therefore by (2.11.2) we conclude that B is uniformly weakly compositumwise unramified over A at Z.

In view of (2.11.3), under the hypothesis of (2.11.5), for i= 1, *.., Y, there exists a subring Ci of B with AC Ci such that B is compositumwise unramified over Cj at Zi, Ci is a finite A-module, and Ci is separable algebraic over A. Upon letting C= A[C,, . . . . C,], in view of (2.9) we see that B is compositumwise unramitied over C at Z, and obviously C is a finite A-module and C is separable algebraic over A. Therefore by (2.11.3) we conclude that B is uniformly almost compositumwise unramified over A at Z.

Remark (2.12) Again let A be a subring of a domain B, and let Zc Z(0, B). We could weaken the concept of uniformly weakly almost

208 ABHYANKARANDHEINZER

unramihedness introduced in (2.8) by saying that B is uniformly weakly semialmost compositumwise unramified over A at 2 if there exists a subring C of B with A c C such that C is almost finite over A, and B is compositumwise unramified over A at 2. Likewise we could weaken the concept of uniformly almost unramifiedness by saying that B is uniformly semialmost compositumwise unramitied over A at 2 if there exists a subring C of B with A c C such that C is separable algebraic and almost finite over A, and B is compositumwise unramified over A at Z.

Note that if A is prepseudogeometric then obviously B is uniformly weakly almost compositumwise unramified over A at Z* B is uniformly weakly semialmost compositumwise unramified over A at Z. Likewise, if A is normal noetherian then B is uniformly almost compositumwise unramified over A at Z-=-B is uniformly semialmost compositumwise unramilied over A at Z. Now in our principal applications of these concepts A will be respectively prepseudogeometric or normal noetherian. Hence the added generality of introducing the above weaker concepts would be rather illusory. Similar remarks apply to the concepts without the uniformity hypothesis.

3. PRELIMINARDES ON GENERALIZED LOCAL RINGS AND IDEALLY CLOSED SUBRINGS, THEOREMS OF COHEN AND EAKIN, AND

LEMMAS OF DEDEKIND AND KRONECKER

In Theorems 2 and 3 of [S] Cohen proved the following two Theorems (3.1) and (3.2), respectively.

THEOREM (3.1) Let R be a glr. Then R is a glr. Zf Q is any ideal in R which is primary for M(R), then Qff is primary for M(Z?) and (Qj) n R = Q. Zf Q’ is any ideal in i? which is primary for M(Z?), then Q’ n R is primay for M(R) and (Q’ n R) l? = Q’.

THEOREM (3.2) A complete glr is noetherian.

As a characterization of ideally closed subrings we have

LEMMA (3.3) Let A be a subring of a ring B such that A is finitely generated ideally closed in B. Then A is ideally closed in B. Zf; moreover, B is noetherian then A is noetherian.

ProoJ Given any ideal J in A and any x E (JB) A A, we can express x as a finite sum x = xlsl + . . . + x,s, with xi E J and si E B; upon letting z= (Xl) . ..) x,) A we have x E (ZB) n A; since Z is a finitely generated ideal in

UNRAMIFIED INTEGRALEXTENSIONS 209

A we get that (ZB) n A = Z; consequently x E Z, and hence x E J. Thus A is ideally closed in Z?. Now assume that B is noetherian; given any sequence of ideals Jr c J, c . ..in A we get a sequence of ideals J,Bc J,Bc --*in B; since B is noetherian, there exists a positive integer n such that J, B = J,B for all m > n; since A is ideally closed in B, we conclude that J, = (J, B) n A = (J, B) n A = J,, for all m > n; therefore A is noetherian.

Concerning closed ideals we obviously have

LEMMA (3.4) For any ideal Z in a glr R we have that Z is closed in R*(ZZ?)nR=Z.

Since the zero ideal in any ring is clearly finitely generated, as an immediate consequence of (3.1) to (3.4) we get

LEMMA (3.5) A quasilocal ring, in which every finitely generated ideal is closed and the maximal ideal is finitely generated, is noetherian.

We note that (3.2) and (3.5) can also be found, respectively, in (31.7) and (31.8) of Nagata [ 131. As a consequence of (3.5) we deduce

LEMMA (3.6) Let S be a quasilocal ring such that M(S) is finitely generated. Assume that there exists a cofnal family (Re)eeE of quasiZocaZ subrings of S such that for every e E E we have that S dominates R,, R, is ideally closed in S and every finitely generated ideal in R, is closed, Then S is noetherian.

Proof In view of (3.5) it suffices to show that given any finitely generated ideal J in S and given any x E nz O (J+ M(S)‘), we have x E J. Since M(S) and J are finitely generated, we can take a finite number of elements yl, . . . . ym, zr, . . . . z, in S such that ( yl, . . . . y,) S= M(S) and (2 1, *.*, z,) S = J. Since the given family is colinal in S, we can find’ e E E such that the elements x, yr, . . . . y,, zi, . . . . z, are all contained in R,. Since S dominates R, and R, contains a set of generators of M(S), we get that M(R,) S=M(S). Let Z= (zi, . . . . z,,) R,. Then Z is a finitely generated ideal in R, and hence (7,“=, (Z+M(R,)‘)=Z. Now IS= J and M(R,) S=M(S); consequently (I+ M(R,)‘) S = J+ M(S)’ for all i; since R, is ideally closed in S, we conclude that (J+ M(S)‘) n R, = Z+ M( R,)’ for all i. Thus

XE fi ((JfM(S)‘)nR,)= 5 (Z+M(R,)‘)=ZcJ. i=O i=O

By Krull’s intersection theorem, every ideal in a local ring is closed; hence by (3.6) we get


LEMMA (3.7) Let S be a quasilocal ring such that M(S) is finitely generated. Assume that there exists a cofinal family (R,), t E of local subrings of S such that for every e E E we have that S dominates R,, and R, is ideally closed in S. Then S is noetherian.

In the situation of (3.7), if M(R,) S=M(S) for some eE E, then obviously M(S) is finitely generated; therefore by (3.7) we get

PROPOSITION (3.8) Let S be a quasilocal ring. Assume that there exists a cofinal family (R,), E E of local subrings of S such that for every e E E we have S dominates R, and R, is ideally closed in S, and such that for some e E E we have M(R,) S = M(S). Then S is noetherian.

In connection with the above Proposition (3.8), but without using it, we now prove

PROPOSITION (3.8*) Let S be a local ring. Assume that there exists a cofinal family (R,),, E of local subrings of S such that for every e E E we have that S dominates R,, R, is ideally closed in S, M(R,) S= M(S), and dim R, = d where d is a nonnegative integer independent of e. Then dimS=d.

Proof Fixing any e E E and remembering that dim R, = d, we can find elements x1, . . . . xd in R, together with a positive integer n such that M(R,)” c (x1, . . . . xd) R,; since M(R,) S = M(S), it follows that hqsy c (Xl) . ..) xd) S, and hence dim S 6 d. Upon letting dim S = q we can find elements yr, . . . . yq in S together with a positive integer t such that M(S)’ = (Y 1, *.., y,) S; since the family (R,),, E is colinal in S, we can find e E E such that the elements y,, . . . . yq belong to R,; since M(R,) S = M(S) and R, is ideally closed in S, it follows that M(R,)’ c (yi, . . . . y,) R, and hence dim R, < q; therefore dim S = q = d.

Remark (3.8A) Note that the conclusion of (3.8*) is not true without the hypothesis of ideally closed. For example, consider the case of a 2-dimensional regular quadratic sequence along an analytic branch. In other words, let S be the valuation ring of a real discrete valuation which birationally dominates and is residually algebraic over a 2-dimensional regular local ring R, and R, is the eth quadratic transform of R, along the analytic branch S. See Abhyankar [l] for details.

As an immediate consequence of the first part of (3.3) we get

PROPOSITION (3.9) Let S be a ring and let R be a subring of S. Assume there exists a cofinal family (R,),, E of subrings of S such that for every e E E we have that R c R, and R is ideally closed in R,. Then R is ideally closed in S.

UNRAMIFIEDINTEGRALEXTENSlONS 211

Here is another lemma about ideally closed subrings.

LEMMA (3.10) Let A be a ring and let B be a subring of the total quotient ring of A with A c B, such that A is nonzerodivisor principal ideally closed in B. Then A = B.

Proof. Given b E B, since B is contained in the total quotient ring of A, we can write b = aJs where a and s are elements of A such that s is nonzerodivisor in A. Therefore a E: (sB) n A = sA because A is nonzerodivisor principal ideally closed in B. Consequently a/s = b E A. Thus A = B.

The following theorem of Cohen [6] can also be found in (3.4) of Nagata [113].

THEOREM (3.11) If every prime ideal in a ring R is finitely generated then R is noetherian.

As a corollary of (3.11) we deduce

LEMMA (3.12) Let R be a quasilocal ring such that M(R) is finitely generated, and let Q be an ideal in R which is primary for M(R). Then Q is finitely generated.

Proof. Since M(R) is finitely generated, we have M(R)” c Q for some positive integer n. Let u : R -+ R/M(R)” be the canonical epimorphism. Then u(M(R)) is the only prime ideal in u(R), and u(M(R)) is finitely generated. Therefore by (3.11) we see that u(R) is noetherian, and hence U(Q) is finitely generated. Since M(R) is finitely generated, so is M(R)“. Therefore Q is finitely generated.

From (3.1) and (3.12) we deduce

LEMMA (3.13) Let I be an ideal in a glr R such that IR is primary for M(R). Then I is primary for M(R).

Proof. Let J= (II?) n R. By (3.1) J is primary for M(R) and hence by (3.12) J is finitely generated. Now JM(R) is primary for M(R) and (JM(R)) I? = (JR)(M(R) I?) = (IR) M(R); consequently by (3.1) we get that (I&) M(R) n R = JM( R). Let any x E J be given; then x G IA and hence we can express x as a finite sum x = xlsl + . . . + x,s, with xi E I and SUE .I?; we can write si= ri+ ti with riE R and tiE M(R); upon letting y=xlr,i- ... -i-x,r, and z=xlt,+ .-. +x,f, we get yfI, x-yE:R, and x-y = z E (I&) M(R); thus x-y E (I&) M(R) n R = JM(R) and y G I, and hence x E I+ JM(R). Thus JC Z-k JM(R); since J is finitely generated, by Nakayama’s lemma we conclude that I= J. Therefore I is primary for M(R).


In the realm of Cohen’s theorem [6], Eakin [S] proved the following theorem which may also be found on p. 263 of Matsumura [12].

THEOREM (3.14) If a noetherian ring B is a finite module over a subring A, then A is noetherian.

For a proof of the following lemma of Kronecker see (1.16) of Abhyankar [2] or (2.27) of Abhyankar [2].

LEMMA (3.15) Let A be a normal domain with quotient field K. Now if f(X), g(X) are manic polynomials in X with coefficients in K such that f(X) = g(X) g*(X) and fX) E A[X], then g(X) E A[X] and g*(X) E A[X]. Hence, in particular, tf x is an element in an overfield of K such that x is integral over A, i.e., such that f(x) = 0 for some manic fX) E A [Xl, then for the minimal manic polynomial g(X) of x over K we have g(X) E A [Xl.

In the above lemma, if d is the degree of g(X), then clearly 1, x, x2, . ..) xd--l is a free A-module basis of A [xl. Thus we get

LEMMA (3.16) Let A be a normal domain and let x be an element of an overdomain of A such that x is integral over A. Then A[x] is a finite-free A-module.

Having cited Kronecker, we must also cite Dedekind. So let us note the following lemma from p. 376 of Dedekind [7] based on Lagrange inter- polation; for a proof also see p. 134 of Hecke [ 111, or line 10 on p. 183 of Abhyankar and Sathaye [4] (where we take the element a in the com- plementary module to be 1).

LEMMA (3.17) Let A be a normal domain with quotient field K. Let x be an element in an overfield of K such that x is integral over A, and let g(X) be the minimal manic poIynomia1 of x over K. Then for every y E K(x) which is integral over A, we have yg’(x) E A[x].

Note that in the above reference to Abhyankar and Sathaye [4], x is assumed separable over K; however, if x is not separable over K, then the assertion is trivially true because in that case g’(x) = 0. Now, in the situation of (3.15), if f(X) E A[X] and f(x) = 0 then f(x) =g(x)g*(x) and f’(x)=g’(x)g*(x)fg(x)g*‘(x)=g’(x)g*(x), and by (3.15) we have g(x) E A[x] and g*(x) E A[x]. Hence we get (3.18) as stated below and in view of (3.17) we also get (3.19) as stated below.

LEMMA (3.18) Let A be a normal domain with quotientfield K. Let x be an element in an overdomain of K such that f(x) = 0 #f’(x) where


fW)EACW . 1s manic. Then for the minimal manic polynomial g(X) of x over K we have g(X) E A [X] and g(x) = 0 #g’(x).

LEMMA (3.19) Let A be a normal domain with quotient field K. Let x be an element in an overdomain of K such f (x) = 0 where f (x) E A[X] is manic. Then for every y E K(x) which is integral over A, we have yf ‘(x) E A[x].

4. ~DERIVATIVEWISE UNRAMIFIED QUASILOCAL RINGS

Let S be a quasilocal ring and let D be a subring of S. We start by proving the following.

PROPOSITION (4.1) Assume that S is elementwise unramij?ed over D. Then S is essentially finite and unramtjied over D.

Proof By assumption there exists x E S and manic f(X) E D[X] such that f(x) = 0, f’(x) $ M(S), and S = D[x] M(SJ n ocx, where the right-hand side is the localization of D[x] at M(S)nD[x]. Now D[x] is a finite D-module and hence S is essentially finite over D.

Let P = M(S) n D. Then P is a prime ideal in D with (PS) n D = P. Let u : S --f SIPS be the canonical epimorphism, let L = u(S), and let K = the quotient held of u(D). Then L is a quasilocal ring, K is a subfield of L, and S is unramified over D *L is a separable algebraic field extension of K.

Let y = u(x), and let g(X) be manic element in K[X] ‘obtained by applying u to the coefficients of f(X). Then g(y) =O, g’(y)+ M(L), and L = 01 M~)nfaYl’ Let v : K[X] -+ K[y] be the unique K-epimorphism with v(X) = y, and let Q = v-‘(M(L) n K[y]). Then Q is a prime ideal an K[X] with g(X] E Q and g’(X) # Q. Now Q is a nonzero prime ideal in K[X] and hence Q = h(X) K[X] where h(X) is a nonconstant manic irreducible element in K[X]. Since g(X)E Q, we can write g(X) = h(X)n q(X) where n is a positive integer and q(X) E K[X] with q(X)+ Q. Now q(X) $ Q implies that q(y) $ M(L) and hence q(y) is a unit in L; also 0 = g(y) = h( y)” q(y), and hence we get h(y)” = 0. We have nh(X)“- ’ h’(X) q(X) + h(X)n q’(X) = g’(X) $ Q = h(X) K[X], and hence we must have n = 1 and h’(X) & Q. Consequently h(y) = 0 and h’(y) # 0; therefore K[y] is a separable algebraic field extension of K. Since L = K[y~M~L~nK~yl and K[y] is a field, we must have L= K[y].

To prove the converse (4.5) of (4.1), we need the following consequence of Nakayama’s lemma.

LEMMA (4.2) Let T be a quasilocal ring and let x be an element in an overring of T such that x is integral over T, and let A = T[x]. Then given


any manic g(X) E T[X] with g(x) E M(T) A, there exists a manic f(X) E T[X] such that f(x) = 0 a&j(X) -g(X) E M( T)[X] (note that in view of the last condition we must have degf(X) = deg g(X)).

Prooj Since x is integral over T, we know that A is a finite T-module. Let d be the degree of g(X) and let B be the T-submodule of A generated by 1, x, . . . . xd- ‘. Since g(X) is manic of degree d, we must have A = B + g(x) A; since g(x) E M(T) A, we now get A = B + M(T) A; therefore A = B by Nakayama’s lemma; consequently upon letting H be the set of all polynomials in X of degree < d with coefficients in M(T), we get M(T) A = {q(x) : q(x) E H}. Since g(x) EM(T) A, there now exists h(X) E H with g(x) = h(x). It suffices to take f(X) = g(X) - h(X).

For proving (4.5) we also need the following two lemmas:

LEMMA (4.3) Let A c B be subrings of S such that S = BMM(sJnB, B is integral over A, and M(S) n B is the only prime ideal in B whose contraction to A is M(S) n A. Then S is the localization of B at A\M(S).

ProoJ Let C be the localization of B at A\M(S) in S. Then s= G(S)nC.

Given any prime ideal P in C let Q = P n B. Then Q is a prime ideal in B with QC= P and Q n (A\M(S)) = @. Now Q n A is a prime ideal in A contained in M(S) n A; therefore by the going up theorem there exists a prime ideal Q’ in B with Q c Q’ and Q’ n A = M(S) n A; by assumption we must now have Q’= M(S) n B. Therefore Q c M(S) n B, and hence P=QCcM(S)nC.

Thus every prime ideal in C is contained in M(S) n C. Therefore C is quasilocal with M(C) = M(S) n C. Since S= CMcSJnC, it follows that s= c.

LEMMA (4.4) Let P be a maximal ideal in a ring A, and let B be an overring of A such that B is a finite A-module. Then there exist only a finite number of distinct prime ideals P,, . . . . P, (r Z 1) in B containing P; P, , . . . . P, are maximal ideals in B; and PB = Q, n . . . n Qr where Qi is an ideal in B which is primary for Pi for I< i < r.

ProoJ: Let v : B + B/PB be the canonical epimorphism. Now B is integral over A, and hence (PB) n A = P. Thus v(A) is a subfield of v(B), and v(B) is a finite v(A)-module. Therefore there exist only a finite number of distinct prime ideals PA, . . . . P, (r > 1) in v(B); PI, . . . . Pr are maximal ideals in v(B); and (0) = QI n . . . n Q, where Qi is an ideal in v(B) which is primary for Pi for 19 i < r. Now it suffrcies to take Pi = V-‘(Pi) and Qi=v-‘(Qi) for 1 <i<r.


We can now prove the following converse of (4.1).

PROPOSITION (4.5) Assume that S essentially finite and unramified over D. Then S is elementwise unramified over D.

Proof: Clearly S is essentially finite and unramified over D iff S is essentially finite and unramified over the localization R = DMcsjn D of D at M(S) A D in S. Similarly, S is elementwise unramified over D iff S is elementwise unramified over R. Hence in proving (4.5) we may pass to In other words, now S dominates the quasilocal subring R, and S is essentially finite and unramified over R. We want to show that S is e~eme~twis~ unramified over R.

Since S is essentially finite over R, there exists a subring B of S such that R c B, B is a finite R-module, and S= BMCSjnB. By (4.4) there are only a finite number of distinct prime ideals PI, . . . . P, (u 3 1) in B ~o~tai~i~~ M(R), and they are ail maximal. Now M(S) R 1% is a prime ideal in B containing M(R), and hence M(S) n B = Pj for some j; upon relabel~i~~ P 1, . . . . P, we may suppose that M(S) A B = P,. Let v : S -+ S/M(S) be the canonical epimorphism. Since S is unramified over R, w is a separable algebraic field extension of v(R); since S = a finite R-module, we see that v(S) = v(B) and u(B) is algebraic field extension of v(R). Therefore there exists 0 # y E v(B) such that u(B) = v(R)[y]. We can take a manic ME R[X] such that upon applying v to the coefficients of h(X) we get the minimal manic ~olyno~~a~ q(X) of y over v(R). Note that then q(y) = 0 #q’(y). Clearl P,S(P,n ... n P,) = B and hence there exists XE P, n . .I n P, wit v(x) =y~ We now have x 4 M(S), h(x) E M(S), and h’(x) $ M(S).

Let pi= Pin R[x]. Since B is integral over R[x are exactly all the prime ideals in R[x] containing maximal (P, , . . . . H, may not be distinct).

Let R’ = RC4MM(sj n RCxl. Then R’ is a quasilocal subring of S such t S dominates R’, and R’ dominates R. Since S is unramified over we know that M(R) S = M(S), and hence M(R’) S = M(S). Since u(S) = v(R)[v(x)], we also have that S is residually rational over R’. N=R[x]\M(S). Now x$M(S)nR[x]=~, and x~H*n ... np hence M(S) n B is the only prime ideal in B whose contraction to R M(S) n R[x]; since B is integral over R[x] and S=BMtSjnB, by (4.3) we now see that S= B,. Since R’= R[xfN and B is a finite R[x]-module, we conclude that S is a finite R’-module; since M(R’) S = M(S) and S is residually rational over R’, by Nakayama’s lemma, we now see that S=

By (4.4) we have M(R) R[x] = 0, n ... n & where Qi is an ideal in R[x] which is primary for i’, for 1 <i< r. Since S= R[x]MM(SjnRlxl and M(S)nR[x]=i?,, we must have &, = P,. Hence in particular


h(x) E oi. Since x E P, n . ‘. A ir,, there exists a positive integer n such that c&&n -.. n &. Let g(X) = Iy7zh(X). Then g(x) = x”h(x) E & n . . . n &? = M(R) R[x]; therefore by (4.2) there exists monicf(X) E R[X] such that

f(x) = 0 and f(X) -g(X) E M(R)[X]. Now g’(x) =nxn-lh(x) + x”h’(x), h(x) E M(S), x # M(S), and h’(x) $ M(S); therefore g’(x) $ M(S). Since f 03 - dW E M(R) CXI, we also have f’(x) -g’(x) E M(S). Therefore f’(x) $ M(S). Thus S . IS e ementwise 1 unramified over R.

In view of (2.9), by (4.1) and (4.5) we get the “only if” part of the following proposition; the “if” part is obvious.

PROPOSITION (4.6) S is finite-compositumwise unramified over D tf and only tf S is elementwise unramt$‘ed over D.

As a consequence of (4.1), (4.5), and (4.6) we get the following characterization of compositumwise unramified extensions.

PROPOSITION (4.7) Let (RJeeE be the family of all quasilocal subrings of S which are dominated by S and which are essentially Jinite and unramified over D.

Then the family (R,),, E coincides with the family of all quasilocal subrings of S which are dominated by S and which are elementwise unramtfied over D. In greater detail, let W= (x E S : x is derivativewise unramtfied for S over D}. Then associated to each XE W there is a unique e(x)E E with R e(x)=DL-&m~~x,. Moreover xc, e(x) gives a surjection of W onto E.

Upon letting s’ be the localization of D[ W] at M(S) n D[ W] in S, we also have that s’ is a quasilocal ring which is compositumwise unramtfied over D, the family (R,),, E is a cofinal family of quasilocal subrings of s’, and s’ dominates R, for each e E E.

Finally, S is compositumwise unramified over D + S = S e the family (R,),.E is cofinal in S.

In the situation of (4.7), in view of (4.1), R, is unramified over D for each e E E; since the family (R,),,c is cofinal in S’, it follows that S is unramilied over D and hence by (2.9), S’ is unramilied over R, for each e E E; by (4.7) S is compositumwise unramified over D and hence by (2.9), S is compositumwise unramilied over R, for each e E E; therefore by (4.7), and as a supplement to it, we get

PROPOSITION (4.8) Let the situation be as in (4.7). Then S is unramifid over D, and for each e E E we have that R, is unramtfied over D, and S is compositumwise unramtfied over R,, and S is unramtfied over R,.

Hence, in particular, tf S is compositumwise unramtfied over D, then S is


amramified over D, and for each e E E we have that S is compositumwise unramified over R,, and S is unramified over R,.

Next we prove the following lemma about ideally closed subrings.

LEMMA (4.9) Let a ring C be essentially free over a subring A. Assume that for every nonunit ideal J in A we have JC J; C (equivalently, each maximal ideal in A is the contraction of some maximal ideal of C). Then A is ideally closed in C.

Proof. Since C is essentially free over A, there exists a multiplicative set N in a subring B of C with A c B such that (every element of N is invertible in C and) C is the localization of B at N (in C) and such that B has a free A-module basis (x,),~ E. Let I be any ideal in A, and let there be given any y E (KY) n A. Now y E IC* yz E IB for some z EN. Since yz E IE, we can write yz = C iexe with i, E Z (where the summation is over all e E E, and where i, = 0 for all except a finite number of e). Since z E B, we can also write z = C a,x, with a, E A; now z E N + z is invertible in C * the ideal C Ca, in C coincides with C. For the ideal J= C Aa, in A we clearly have JC = C, and hence by our hypothesis we must have J= A.

Now C yaex, = yz =C iex, and hence ya, = i, for all e. Therefore y(C Aa,) c C Ai, c I. Since 2 Aa, = A, we get y E I;

By taking A = D in (3.16) and (3.19) we get

LEMMA (4.10) If S is a domain, D is a normal domain, and S is elementwise unramified over D, then S is normal and S is essentially finite- free over D.

Recall that if N is a multiplicative set in a domain C, and CN is the localization of C at N in an overlield of C, then Q I-+ QC, gives an inclusion preserving bijection of Z*(O, N, C) onto Z(0, C,) and its inverse is given by 4 H q n C; for instance see Corollary 1 on p. 224 of Zariski and Samuel [15]. In view of the going up and going down theorems (for instance see 1.24 and 1.24B of Abhyankar [2]), and in view of the said bijection, we get

LEMMA (4.11) If R* is a normal quasilocal domain, and S* is a quasilocal domain such that S” dominates R*, and S* is essentially integral over R*, then dim S* = dim R*.

Next we prove the following

LEMMA (4.12) Upon letting R be the localization of D at M(S) A D in S, we have &he following.


(4.12.1) If S is elementwise unramtjiied over D, then S is essentially integral over R.

(4.12.2) More generally, if S is almost compositumwise unram$ed over D, then S is essentially integral over R.

(4.12.3) If S is elementwise unramtfied over D, then S is essentially finite over R.

(4.12.4) If S is elementwise unramt$ed over D, and R is noetherian, then S is noetherian.

(4.12.5) If S is elementwise unramtfied over D, S is a domain, and R is normal, then S is normal, S is essentially finite-free over R, R is ideally closed in S, and dim S = dim R.

(4.12.6) If S is almost compositumwise unramtfied over D, S is a domain, and R is normal, then dim S = dim R.

Proof: Parts (4.12.1), (4.12.2), and (4.12.3) are obvious. By (4.12.3) we get (4.12.4), By (2.9), (4.9), (4.10), (4.11), and (4.12.1) we get (4.12.5), By (4.11) and (4.12.2) we get (4.12.6).

As the second supplement to (4.7) we now prove the following

THEOREM (4.13) In the situation of (4.7), assume that S is a domain, and R is normal where R is the localization of D at M(S) n D in S.

Then s’ is normal, R is ideally closed in s’, and dim s’ = dim R, and for every e E E we have that R, is normal, R, is essentially finite-free over R, R is ideally closed in R,, dim R, = dim R, and R, is ideally closed in s’.

Hence, in particular, tf S is compositumwise unramtjiied over D, then S is normal, R is ideally closed in S, and dim S = dim R, and for every e E E we have that R, is ideally closed in S.

Proof By (4.12.5) we see that for every e E E we have that R, is normal, R, is essentially finite-free over R, R is ideally closed in R,, and dim R,= dim R. By (4.7) we know that S is a cofinal union of the family (U.E and hence by (3.9) we see that R is ideally closed in S, and since a cotinal union of normal domains is obviously normal, we also see that S is normal. By (4.12.6) we also have dim S = dim R.

By (4.7) we know that if S is compositumwise unramified over D then S= S’, and therefore by the above paragraph we conclude that if S is compositumwise unramified over D then S is normal, R is ideally closed in S, dim S = dim R, and for every e E E we have that R, is ideally closed in S.

Thus (assuming that S is a domain and R is normal) we have shown that if S is compositumwise unramified over R then R is ideally closed in S. Given any e E E, we have also shown that R, is normal; now S is a domain


and, in view of (4.8), s’ is compositumwise unramified over R,; so we conclude that R, is ideally closed in S’.

Again by (4.7) we know that if S is compositumwise unramified over D then S = S’, and therefore by the above paragraph we conclude that if S compositumwise unramified over D then for every e E E we have that R, is ideally closed in S.

As the main supplement to (4.7) we now prove the following

THEOREM (4.14) In the situation of (4.7), assume that S is a domain and R is a normal noetherian domain where R is the localization of D at M(S) A D in S.

Then S’ is a normal noetherian domain with dim s’ = dim R, and for every e E E we have that R, is a normal noetherian domain with dim R, = dim R.

Hence, in particular, if S is compositumwise unramified over D, then S is a normal noetherian domain with dim S= dim R.

Proof By (4.8) and (4.13) we see that S’ is a normal domain with dim s’ = dim R, and for every e E E \rre have that R, is a normal domain with dim R,= dim R. By (4.7), (4.8), (4.12.4), and (4.13) we see that S’ is a quasilocal ring which is a cofinal union of the family (RJeEE, and for every e E E we have that R, is a local ring such that S dominates R,, R, is ideally closed in S’, and M(R,) S = M(S); hence by (3.8) we see that S is noetherian.

By (4.7) we know that if S is compositumwise unramihed over D then S = S’, and therefore by the above paragraph we conclude that S is a normal noetherian domain with dim S = dim R.

Remark (4.14A) For a result related to (4.14) see (10.3.1.3) of OIII of Grothendieck [9].

Finally, as a consequence of (4.7), (4.13), and (4.14) we prove

THEOREM (4.15) In the situation of (4.7), we have the following.

(4.15.1) If S is a domain and D is prequasipseudogeometric, then S is prequasipseudogeometric. Hence, in particular, tf S is a domain, D is prequasipseudogeometric, and S is compositumwise unramified over D, then S is prequasipseudogeometric.

(4.15.2) If D is quasipseudogeometric, then S is quasipseudogeometric. Hence, in particular, tf D is quasipseudogeometric, and S is compositumwise unramified over D, then S is quasipseudogeometric.

(4.153) If S is a domain and D is a normal prepseudogeometric domain, then S is a normal prpseudogeometric domain. Hence, in particular, if S is a domain, D is a normal prepseudogeometric domain, and S is’ com-

481/136/l-15


positumwise unramified over D, then S is a normal prepseudogeometric domain.

(4.154) If S is a domain and D is a normal pseudogeometric domain, then s’ is a normal pseudogeometric domain. Hence, in particular, if S is a domain, D is a normal pseudogeometric domain, and S is compositumwise unramified over D, then S is a normal pseudogeometric domain.

ProoJ: To prove (4.15.1), for a moment assume that S is a domain and D is prequasipseudogeometric. Given any finite algebraic field extension L* of the quotient field L’ of s’, we want to show that the integral closure of s’ in L* is a finite S-module. Now clearly there exists a finite algebraic field extension K* of the quotient field K of D such that L* = L(K*). Let D* be thejntegral closure of D in K*. Then D* is a finite D-module because D is assumed to be prequasipseudogeometric. Let C= S’[D*]. Then clearly C is a finite S-module, C is integral over s’, and L* is the quotient field of C. We shall show that C is normal, and that will complete the proof of the assertion that S is prequasipseudogeometric. By taking localizations in L* we have C = 0 eE Mz(D,cj C,. Therefore, to show that C is normal, it suffices to prove that, given any Q EMZ(O, C), upon letting S* = C, we have that S* is normal. Now (say by 1.19 of Abhyankar [21]) D* dominates S, and hence S* is the localization of D*[ W] at M(S*) n D*[W], and W is derivativewise unramified for S* over D*. Therefore S* is compositumwise unramified over D* and hence by (4.13) we conclude that S* is normal. Thus S is prequasipseudogeometric, and hence in view of (4.7) we also see that if S is compositumwise unramified over D then S is prequasipseudogeometric. This completes the proof of (4.15.1).

To prove (4.15.2), let us drop the assumptions of S being a domain and D being prequasipseudogeometric, but let us assume that D is quasipseudogeometric. Given any prime ideal Q in S’, upon letting u : S + S/Q be the canonical epimorphism, we see that the quasilocal domain u(S) is the localization of u(D)[u( W)] at M(u(S’)) n u(D)[u( W)] in u(S’), and u(W) is derivativewise unramified for u(S) over u(D). Thus u(S’) is compositumwise unramified over u(D), and by assumption u(D) is prequasipseudogeometric. Therefore by (4.15.1) we see that u(S’) is prequasipseudogeometric. This being so for every prime ideal Q in S’, we conclude that S is quasipseudogeometric. Therefore by (4.7) we also see that if S is compositumwise unramified over D then S is quasipseudogeometric. This completes the proof of (4.15.2).

Part (4.15.3) follows from (4.14) and (4.15.1). Similarly, (4.15.4) follows from (4.14) and (4.15.2).

Remark (4.15A) For results related to (4.15) see Greco [lo].


5. RESIDUE CLASS RJNGS MODULO A UNIVARIATE MONIC POLYNOWAL

Let B be a ring and let A be a subring of B. We shall now give a string of remarks and lemmas culminating in Theorems (5.14) and (5.15) in which we shall give sufficient conditions for a prime ideal in B to be the extension of its contraction in A.

Remark (5.1) Given any ring homomorphism u : B -+ B*, and any ideal Z in B with Ker u c Z, upon letting B= U(B), d = u(A), 7= u(Z), P = Z n A, and P = fn k Then, by well-known properties of epimorphisms, we get the following.

-- (5.1.1) Q F--+ u(Q) gives a bijection of Z(Z, B) onto Z(Z, B); its inverse

is given by Q H u-‘(Q). The said bijection induces a bijection of mZ(Z, B) -- onto mZ(Z, B). It also induces a bijection of Z(Z, B; P, A) onto Z(z B; p, A). Hence in paticular we have that

-- -- Z(Z, B; P, A) = mZ(Z, B) e Z(I, B; P, A) = mZ(Z, B).

(5.1.2) If mZ(Z, B)= (Q,, . . . . Qr} = a Iinite set of cardinality r, and Z= HI n ... n H, where, for i= 1, . . . . Y, the ideal Hi in B is primary for Qi, then I= u( H,) n . . . n u(H,) where, for i = 1, . . . . Y, the ideal u(HJ in B is primary for u(Qi), and moreover:

(1) for any ye:B and i~{l,...,~) we have that JJ$Q* 4~) # MQJ;

(2) for any i E ( 1, . . . . r> we have that Hi=Qiou(H,)=u(QJ; and (3) for any ifs { 1, . . . . r> we have that B is residually separable

algebraic over A at Qio i? is residually separable algebraic over A at 4QJ

(5.1.3) --

If mZ(Z, B)= @,, . . . . Q,} = a finite set of cardinality r, and T=R,n . . . n ir, where, for i = 1, . . . . r, the ideal Bi in B is primary for Qi, thenZ=u-i(A,)n . . . nu- ‘(il,) where, for i= 1, . . . . r, the ideal U-‘(Z?,) in B is primary for U-‘(Qi).

Remark (5.2) Let N be a multiplicative set in B, and let j : B -+ C a localization map of B at N. Let Z be an ideal in B, and let P = Z n A and Z =j-‘(j(Z) C). Then, in view of the following observation (5.2.1), by the basic properties of localization given on pp. 223 to 227 of Zariski and Samuel [ 151, we get the following assertions (5.2.2) to (5.2.10).

(5.2.1) By Zom’s lemma we see that every member of Z(Z, B) contains some member of mZ(Z, B); i.e., given any Q E Z(Z, B), there exists Q* emZ(Z, B) such that Q* c Q. It follows that every member of Z*(Z, N, B) contains some member of mZ*(Z, N, B); namely, given any

222 ABHYANKAR AND HJSNZER

Q G Z*(I, N, B), by what we have just said, there exists Q’ E mZ(1, B) such that Q’ c (2, and clearly we must have (2’ E mZ*(I, N, B).

(5.2.2) We have I’ = the isolated component of I at N in B, and we have Z(I’, B) = Z*(I, N, B). Moreover, Q w-j(Q) C gives a bijection of Z(rl, B) onto Z(j(1) C, C), and its inverse is given by 4 -j-l(q). The said bijection is inclusion preserving and hence it induces a bijection of mZ(I’, B) onto mZ(j(1) C, C).

(5.2.3) If mZ(I, B)= (Q,, . . . . Q,} = a finite set of cardinality Y and Z=H,n . . . n H, where, for i= 1, . . . . r, the ideal Hi in B is primary for Qi, then upon letting 1 ,< d(l) < ... <d(s) < r be the unique sequence such that NnQj=@ for all iE {d(i), . . . . d(s)}, and NnQi#@ for all iE (1, . . . . r}\{d(l), . . . . d(s)}, and upon letting qi =i(Qdci,) C and hj=j(H,,,,) C for i= 1, . . . . s, we have mZ(j(P), C) = {ql, . . . . qs) = a finite set of cardinality s, andj(1) C= h, n . .. n h, where, for i= 1, . . . . S, the ideal hi in C is primary for qi, and moreover:

(1) for any yEB and in (1, . . . . s> we have that Y $ Qd(i) -j(y) 4 qi; (2) for any i E ( 1, . . . . S} we have that H+, = Qdci) = hi = qi; and (3) for any iE (1, . . . . S} we have that B is residually separable

algebraic over A at Qdci) o C is residually separable algebraic over j(A) at qi.

(5.2.4) If I= I’, and mZ(j(1) C, C) = (ql, . . . . qs} = a finite set of cardinality s and j(1) C = h, n . . . n h, where, for i = 1, . . . . S, the ideal h, in C is primary for qi, then upon letting Q,=j-‘(4,) and H,=j-l(h,) for i=l , . . . . s, we have mZ(I, B) = {Q, , . . . . Q,] = a finite set of cardinality s, and I= Hin ... n H, where, for i= 1, . . . . s, the ideal Hi in B is primary for Qi, and moreover:

(1) for any yeB and icz (1, . . . . S> we have that u~Qi~j(y)~qi; (2) for any i e (i, . . . . S} we have that H, = Qi+ hi = qi; and (3) for any ifc (1, . . . . S} we have that B is residually separable

algebraic over A at Qi o C is residually separable algebraic overj(A) at qi. (5.2.5) By (5.2.1) and (5.2.2) we see that if Z(I, B; P, A) = mZ(1, B)

then Z(I’, B; P, A) = mZ(r, B), and if also Z’ #B then Z’ n A = P E Z(0, A). Note that if Z(I, B; P, A) = mZ(I, B) and I# B then obviously P E Z(0, A).

(5.2.6) If Z= Z’, and if C is an overring of B such that every element of N is invertible in C, and if C is the localization of B at N in C, and j: B+ C is the canonical injection, then by (5.2.2) we see that Z(Z, B; P, A) = mZ(I, B) o Z(ZC, c; P, A) = mZ(ZC, C).

(5.2.7) More generally, if I= I’ then (without any assumption on C) we have that Z(I, B; P, A) = mA(I, B) o Z(j(I) C, C; j(P), j(A)) =


mZ(j(1) C, C). [To see this we note that always Kerj c I’; hence, assuming that I= r, we now have Kerjc c thereforej(P) =j(l) nj(A) and by (5.1.1) we get that Z(I, B; P, A) = mZ(1, B) - Z(j(l), j(B);j(P), j(A)) = mZ(j(I),j(B)). On the other hand, j(N) is a multiplictive set in the ring j(B), every element of j(N) is invertible in the overring C of j(B), C is the localization ofj(B) at j(N) in C, and the assumption I= I’ also implies that for the ideal j(1) in j(B) we have j(1) = (j(1) C) n)(B); consequently by applying (5.2.6) to the quintet j(A), j(B), j(1), j(N), C we get that Z(j(l), j(B); j(P), j(A)) =mZW), AB)~WU) C CAP), j(A)) =m-WU) C, Cl.1

(5.2.8) If Z(I, B; P, A) = mZ(1, B) then Z(j(1) C, C;j(P), j(A)) = mZ(j(1) C, C), and if also j(1) C# C then PE Z(0, A) and (j(1) C) n j(A) =j(P) E Z(0, j(A)). [To see this, since it is obvious when j(l) C= C, we may suppose that j(1) C# C; now I’# B, and assuming that Z(Z, B; P, A) = mZ(1, B), by (5.2.5) we get Z(rl, B; P, A) = mZ(I’, B) and I’ n A = PE Z(0, A). For the ideal I’ in B we always have j(P) C = j(d) C and I’=j-‘(j(r) C), and hence by the * part of (52.7) we get Z(j(1) C, C;j(P), j(A)) = mZ(j(1) C, C). Therefore, since j(1) Cf C, there exists q E Z(j(Z) C, C; j(P), j(A)) an d we now get j(P) c (j(l) C) nj(A) c q n j(A) = j(P) and hence (j(Z) C) n j(A) =j(P) E Z(0, j(A)).]

(5.2.9) If PB=Z and Z(Z, B; P,A)=mZ(Z, B)= {Ql ,..., Q,>= a finite set of cardinality r, and Z= H, n ... n H, where, for i = 1, . . . . r, the ideal Hi in B is primary for Qi, then for any ig’(l, ..,, Y) we have that B is unramified over A at Qio Hi = Qi and B is residually separable algebraic over A at Qi. [This follows from (5.2.3) and (5.2.8) by taking N= B\Qi.]

(5.2.10) If PB=Z and Z(Z,B;P,A)=mZ(Z,B)={Q, ,..., QF>= a finite set of cardinality r, and Z= H, n . . . n H, where, for i= 1, . . . . r, the ideal Hi in B is primary for Qi, then, with S, qi, hi as in (5.2.3), for any its { 1, . . . . S> we have that C is unramified over j(A) at qiohi= qi and C is residually separable algebraic over j(A) at qi, [This follows from (5.23) and (5.2.9) by noting that if j* : C -+ C* is a localization map of C at q1 then the composition j’ : B -+ C* of j and j* is a localization map of B at Qd(ij.1

Remark (5.3) Let A” be a domain with quotient field K. Given any nonconstant manic polynomial g(X) E A”[X] = i3, let g(X) = &(xp. * .&(X) w be the factorization of g(X) in K[X] = c where Y, t(1), me.9 U(r) are positive integers and El(X), . . . . g,(X) are pairwise distinct nonconstant manic irreducible polynomials in X with coefficients in K. Let r=g(X) B and let H be the zero ideal in ji. For i= 1, . . . . r, let qi= gi(X) c’, K,=g,(X)‘(“)e, Qi=gin& and Ri=hin8. Given any g*(X)e;i[X] let m = ax) g*(x).


(53.1) Obviously Z(re, c; P, A”) =mZ@‘, c)= {gl, . . . . q,> = a finite set of cardinality I, and 72; = h”, A . . . n z, where, for i = 1, . . . . Y, the ideal & in 2; is primary for qi. Given any ie { 1, . . . . r], upon letting t = t(i) and G(X) = g,(X) and

G*(X) =&(J-)t(l). . .gi-l(X)t(i-l)gi+l(X)t(i+l). . .&5(X)‘(‘I

we have

g’(X) = tG(X)‘-’ G’(X) G*(X) + G(X)’ G*‘(X)

and hence $(x)$#4i*t=1 and G’(W # 4i

and clearly

and

e G’(X) # 0

e c is residually separable algebraic over K at gi

e 2; is residually separable algebraic over A” at qi.

Thus, for any ie { 1, . . . . r} we have that g’(X) # gio& = qi and c is residually separable algebraic over A”. Obviously y’(X) = g’(X) g*(X) + g(X) g*‘(X) and hence for any i E (1, . . . . r > we also have that T(X) $ gj * iv3 # 4i.

(5.3.2) Now clearly p= r”n A”, and I”= (72;) n 8, and c= the localization of B at A”\(O) in the quotient field of B. Therefore, in view of (5.2.4) and (5.2.6), by (5.3.1) we see that Z(z 8; i7, A”) = mZ(7, B) = (Q1, . . . . QI} = a finite set of cardinality r, and I” = r?, n . . . n I?,. where, for i= 1, . . . . r, the ideal iri in B is primary for ei, and for any iE (1, . . . . Y} we have that y(X) 4 &=>g’(X) 4 &-s I??z = Qi and B is residually separable algebraic over A” at ei.

Remark (5.4) Assume that A is a normal domain, B is a domain, and B= A[x] with XE B such that f(x) = 0 where f(X) E A[X] is a nonconstant manic polynomial. Let g(X) be the minimal manic polynomial of x over the quotient field of A. Now by (3.15) we see that g(X)E A[X] is a nonconstant manic polynomial, and f(X) =g(X) g*(X) with g*(X) E A[X]. Also clearly Ker v =g(X) A[X] where v : A[X] --f A[x] is the unique A-epimorphism with v(X) = x, and clearly f’(x) =g’(x) g*(x) with f’(x), g’(x), g*(x) in B. Given any prime ideal P in A, let

UNRAMWIED INTEGRAL EXTENSIONS 225

u : B -+ B/(PB) be the canonical epimorphism, let A” = u(A) and % = u(x), and let T(X), g(X), g*(X) be the polynomials in X with coefficients in A” obtained by applying u to the coefficients of f(X), g(X), g*(X), respectively. Note that now obviously u(B) = A”[Z], g(X) is a nonconstant manic polynomial, y(X) =g(X) g*(X), and Ker ii =g(X) B where zi : B = J[X] -+ A”[Z] is the unique A”-epimorphism with z?(X) = E. Also note that by the lying over theorem (see 1.20 of Abhyankar [2]) we have (Ker U) n A= P and hence A” is a domain. Let r, Qi, Bi be as in the preamble of (5.3). For i = 1, . . . . r, let Qj=u-‘(ii(Qi)) and III~=u-~(B(W,)).

Now in view of (5.3.2) and (5.2.9), by applying (5.1) first to E and then to u (or, alternatively, first to u” and then to v, where v” : A[XJ -+ A”[X] is the unique epimorphism such that i?(z) = u(z) for ail z E A, and v”(X) = X), we see that Z(P, B; P, A) = mZ( P, B) = (Qr , . . . . QP > = a finite nonempty set of cardinality Y, and PB = H, n - . . n H, where, for i = 1, . . . . r, the ideal H, in B is primary for Qj, and for any in (1, . . . . r> we have that f’(x) $ Qi *g’(x) $ Qj * Hi = Qi and B is residually separable algebraic over A at Qi e B is unramified over A at Qi.

Therefore, in view of,(5.2.3), (5.2.8), and (5.2.10), we see that, given any multiplicative set N in B, upon letting C be the localization of B at Jv in the quotient field of B, and upon letting 1 d d(l) < . . . < d(s) < r be the unique sequence such that Nn Qi= @ for all itz (d(l), . . . . d(s)), and NnQi#@ for all ie (1, . . . . r}\(d(l), . . . . d(s)}, and upon letting qi= (Qd(i)) C and hi=(H~~i~) C for i= 1, . . . . S, we have Z(P, C; P, A) = mZ(P, C) = (41, *.., 4s) = a finite set of cardinality s, and PC- bin ..- nh, where, for i= 1, . . . . s, the ideal hi in C is primary for qi, and for any i,E (1, . . . . s>, we have that f’(x)#qi*g’(x)$qi~hi=qi and C is residually separable algebraic over A at qie C is unramified over A at qi.

Remark (5.5.) To generalize (5.4) somewhat, assume only that A and B are rings and B= A[x] with XE B such that Ker 1; =g(X) A where Y : A[X] + A[x] is the unique A-epimorphism with t;(X) =x, and g(X) E A[X] is a nonconstant manic polynomial. Given any g*(X) E A[X] let f(X) =g(X) g*(X). N ow clearly f’(x) = g’(x) g*(x) with S’(x), g’(x), g*(x) in B. Given any prime ideal P in A, let u : B + B/(PB) be the canonical epimorphism, let A” = u(A) and 3 = u(X), and let j’(X), g(X), g*(X) be the polynomials in X with coefficients in A” obtained by applying u to the coefficients of f(X), g(X), g*(X), respectively. Note that now obviously u(B) = ACT], g(X) 3w =im if*Gn

is a nonconstant manic polynomial, and Ker ii =2(X) B where ii : 3 = A[.Y] 4 A[Z] is the

unique A”-epimorphism with ii(X) =Z. Also note that by the lying over theorem (see 1.20 of Abhyankar [2]) we have (Ker U) n A = P and hence A” is a domain. Let r, Qi, Ri be as in the preambie of (5.3). For i = 1, . . . . r, let Qi=u-‘(ii(Qi)) and Hi=u .‘(ii(fi,)).


Now in view of (5.3.2) and (5.2.9), by applying (5.1) first to ii and then to u (or, alternatively, first to v” and then to v, where u” : A[X] -+ A”[X] is the unique epimorphism such that v”(z) = U(Z) for all z E A, and v”(X) = X), we see that Z(P, B; P, A) = m.Z(P, B) = { Qr, . . . . Q,} = a finite nonempty set of cardinality I, and PB = H, n ... n H, where, for i= 1, . . . . r, the ideal Hi in B is primary for Qi, and for any ie (1, . . . . r} we have that f’(x)$Qi*g’(x)$Qi-+Hi=Qi and B is residually separable algebraic over A at Qi* B is unramified over A at Qi.

Therefore, in view of (5.2.3), (5.2.8), and (5.2.10), we see that, given any multiplicative set N in B, upon letting j : B + C be any localization map of B at N, and upon letting 1 < d( 1) < . . . < d(s) < r be the unique sequence such that Nn Qi = @ for all iE {d(l), . . . . d(s)), and N n Qi # @ for all iE { 1, . . . . rl\{d(l), . . . . d(s)), and upon letting qi=j(Q& C and hi= j(H,& for i = l., . . . . s, we have Z(j(P), C;j(P),j(A)) = mZ(j(P), C) = j:; ,r z p} = a fintte set of cardinality s, and j(P) C = hl n . . . n h, where,

5 .*., s, the ideal hi m C is primary for qi, and for any ie { 1, . . . . s}, upon letting j(f)(X) and j(g)(X) be the polynomials in X with coefficients in j(A) obtained by applying j to the coefficients of f(X) and g(X), respectively, we have that j(f)'(x) $ qi *j(g)’ (x) # qi 0 hi = qi and C is residually separable algebraic over j(A) at qio C is unramified over j(A) at qi.

Remark (5.6). Let I be an ideal in B. Note that for every Q E mZ(I, B) we have that the isolated component j*[I, Q, B] is an ideal in B which is primary for Q (for instance see 1.15 of Abhyankar [a]). Recall that an irredundant primary decomposition of I in B is an irredundant representation I = H, n . . . n H, where H, , . . . . H,. are a finite number of primary ideals in B whose radicals are pairwise distinct prime ideals in B; here irredundant means that none of the Hi can be dropped from the intersection. This representation is unique means if I= J1 n . .. n J, is any other such then s = r and, after a suitable relabelling, Ji = Hi for all i. It can be seen that if I has a unique irredundant primary decomposition I= H, n . . . n H, in B, and if B is noetherian, then upon letting Qi= the radical of Hi in B we have that Q,, . . . . Qr are exactly all the distinct members of mZ(I, B) and Hi =j*[I, Qi, B] for all i. Conversely, without assuming B is noetherian, if mZ(1, Z) is a finite set {Q,, . . . . Q,> of cardinality r, and upon letting Hi =j * [I, Qj, B] for all i we have that I = H, n . . . n H,, then this is an irredundant primary decomposition of I in B and it is the unique such representation. It follows that the decompositions of PB and PC given in (5.4), and the decompositions of PB and j(P) C given in (5.5), are unique irredundant primary decompositions.

As an immediate consequence of (5.4), by assuming P to be unsplit in C and then taking B to play the role of C, we get


LEMMA (5.7) Assume that B is a domain, and A is normal. Let P and Q be prime ideals in A and B, respectively, such that Q is the only prime ideal in B which contracts to P in A. Assume that B is elementwise unramtfied over A at Q. Then PB = Q.

Let us recall that if N is a multiplicative set in a domain C, and C, is the localization of C at N in an overtield of C, then Q H QC, gives an inclusion preserving bijection of Z*(O, N, C) onto Z(0, C,) and its inverse is given by 4 w 4 n C; for this bijection see Corollary 1 on p. 224 of Zariski and Samuel [15]; we may call this the natural correspondence between Z*(O, N, C) and Z(0, C,). As consequences of the going down theorem (see 1.24B of Abhyankar [2]), and the said natural correspondence, we shall now prove the following two lemmas:

LEMMA (5.8) Assume that B is a domain, and B is essentially integral over A. Let P and Q be prime ideals in A and B, respectively, such that Q is the only prime ideal in B which contracts to P in A. Assume that A is normal at MZ(P, A). Then Z(P, B) = Z(Q, B) and MZ(P, B) = MZ(Q, B). Moreover, given any Q’ E Z(Q, B), upon taking localizations in the quotient field~fB,andlettingS=B,~,&=QS, P’=Q’nA,D=A..,andH=PD, we have that S is a quasilocal domain, D is a normal quasilocal domain, S dominates D, i? and Q are prime ideals in D and S, respectively, Q is the only prime ideal in S which contracts to H in D, and (PB) S = 7;s; finally, tf B is compositumwise unramified over A at Q’ then S is compositumwise unramified over D.

Proof Since P c Q, we obviously have Z(Q, B) c Z(P, B). Conversely, given any prime ideal Q* in B with P c Q*, we want to show that Q c Q*. By assumption there exists a subring C of B and a multiplicative set N in C such that A c C, C is integral over A, and B is the localization of G at N in an overfield of B. Let P* = Q* n A. Then P* is a prime ideal in A with PC P*, and hence, in view of the going down theorem and natural correspondence between Z*(O, N, C) and Z(0, B), there exists a prime ideal Q** in B such that Q** c Q* and Q** n A= P. The uniqueness of Q now implies that Q* * = Q. Thus Z(P, B) = Z(Q, B). Therefore MZ(P, B) = MZtQ!, B).

Wow given any Q’ E Z( Q, B), upon taking localizations in the quotient field of B, and upon letting S=B,,, Q=QS, P’=Q’nA, D=AF, and P= PD, obviously S is a quasilocal domain, D is a normal quasilocal domain, and, in view of the natural correspondence between Z(0, Q’, and Z(0, S) and the natural correspondence between Z(0, P’, A) a Z(0, D), we see that S dominates D, P and Q are prime ideals in D and S, respectively, Q is the only prime ideal in S which contracts to p in

228 ARHYANJCAR AND HEINZER

and (PB) S= PS; finally, if B is compositumwise unramified over A at Q’ then obviously S is compositumwise unramified over D.

LEMMA (5.9) Let RO be a normal quasilocal domain. Let S be a quasilocal domain such that S dominates RO, and S is essentially integral over RO. Let & be a prime ideal in S. Assume that there exists a subring D of RO such that upon letting p = Q n D we have that Q is the only prime ideal in S which contracts to p in D. Then Q n RO is the only prime ideal in R, which contracts to i” in D.

ProoJ: By assumption there exists a subring C of S with R,, c C such that C is integral over R,, and S is the localization of C at M(S) n C in S. Let PO be any prime ideal in RO such that PO n D = P. Now P,, c M(R,) = M(S) n RO, and hence by the going down theorem and the natural correspondence between Z*(O, M(S) n C, C) and Z(0, S), there exists a prime ideal Q, in S such that Q0 n R, = P2. Clearly Q0 n D = P,, n D = p = & n D, and hence by the uniqueness of Q we get Q, = 0, and therefore we must have PO = Q n Rd.

Next we prove the following

LEMMA (5.10) Let S be a quasilocal domain, and let D be a normal subdomain of S such that S is compositumwise unramtjied over D. Let H and Q be prime ideals in D and S, respectively, such that Q is the only prime ideal in S which contracts to P in D. Then HS= &, and S is normal.

Prooj By (4.7) and (4.13) we see that S is normal, and S is the cohnal union of a family (R,),, E of normal quasilocal domains R, which contain D and are dominated by S. For each e E E, by (5.9) we see that Q n R, is the only prime ideal in R, which contracts to a in D, and hence by (5.7) we get FR, = & n R,. Since S is the cotinal union of the family (R,),, E, it follows that BS= 0.

Now we can prove the following generalization of (5.10).

LEMMA (5.11) Assume that B is a domain. Let 2 be a subring of A. Let P and Q be prime ideals in 2 and B, respectively, such that Q is the only prime ideal in B which contracts to P in A. Let P = Q n A. [Note that then P is a prime ideal in A, and moreover Q is the only prime ideal in B which contracts to P in A.] Assume that there exists JcA such that B is compositumwise unramified over A at MZ(Q, B) n MZ(J, B), A is normal at MZ(P, A), and for every Q’ E MZ(Q, B)\MZ(J, B) we have PBej = QB,! (we are taking localizations in some overtield of B). Then PB = Q, and B is normal at Z(Q, B) n Z(J, B).


Proqfi Given any Q’ EMZ(O, B), first if Q # Q’ then by (5.8) we would have P Sr: Q’ and hence we would get (PB) B,, = B,, = QBo,, and second if Q c Q’ and J $ (2’ then by assumption we would have QB,, = PBaJ c (PB) B,, c QBo, and hence we would get (PB) B,, = QBo,, and third if Q c Q’ and J c Q’ then upon letting S = Bo, Q = QS, P’ = (2’ n A, D=A,,, and p= PD, by (5.8) and (5.10) we would see that (PB) B,, = (PB) S = BS = Q = QS = QB,,, and S is normal. Thus for every Q’ E MZ(0, B) we have (PB) B,, = QB,,; now for any ideal I in B we have I= (7ewz(~, B) VB,,), and hence we get PB= Q. Since for any Q’ E M.Z(Q, B) n MZ(J, B) upon letting S = B,. we have that S is normal, it follows that B is normal at MZ( Q, Bj n MZ( J, B), and hence B is normal at Z( Q, B) n Z( J, B).

Next, as a consequence of (5.11) we prove the following.

LEMMA (5.12) Assume that B is a domain. Let A be a subring of A. Let P and Q be prime ideals in A and B, respectively, such that Q is the only prime ideal in B which contracts to P in A. Let P = Q n A. [Note that then P is a prime ideal in A, and moreover Q is the only prime ideal in B which contracts to P in A.] Assume that B is essentially integral over A, and PB, = QBe (we are taking localizations in some overfield of B). Also assume that there exists x E A and JC A such that B is composit~mwise unramified over A at MZ(Q, B) n MZ(J, B), A is normaI at MZ(P, A), A - - - - is normal at Z(P, A)\Z(J, A), A is noetherian at Z(P, A)\Z(J, A), for every - - P’ E Z( P, A)\Z(J, 3) we have x2,, = PJpr, B is normal at MZ(Q, B)\ MZ(J, B), and B is noetherian at MZ(Q, B)\MZ(J, B). Then PB = (2.

Prooj In view of (5.11) it suffices to prove that, given any Q’ E MZ(Q, B)\MZ(J, B), we have PB,< = QBo,. Let S = B,,, Q = QS, P’=Q’n& and R=A,. Then clearly P’ E Z(P, A)\Z(J, A) and hence by assumption R and S are normal local domains such that S dominates an is essentially integral over R, and what we have to show is that xS= Q. Since Q is the only prime ideal in B which contracts to i’ in 6, in view of the natural correspondence mentioned just before (5.8), we see that xR = FR = a prime ideal in R, and Q is the only prime ideal in S which contracts to xR in R. If x =0 then, in view of the said natural correspondence, by (1.23) of Abhyankar [2] we would get Q = (0) = xS. So henceforth assume that x # 0. By assumption PBi? = QBB, and hence by a theorem of Krull we see that xR is a prime ideal of height one in R, and xS=DnQ,n ... n Q, where r is a positive integer and Qz, . . . . Qr are primary ideals in S whose radicals Q2, . . . . &, are pairwise distinct mhne ideals of height one in S with Qi # & for i = 2, . . . . r. For i = 2, . . . , r, in view of the said natural correspondence, by the going down theorem (see I.243


of Abhyankar [2]), we see that Qi n R is a prime ideal of height one in R; now since xR c &in R, we get xR = Qi. Therefore we must have Y = 1, i.e., xs=Q

As an easy consequence of (4.14) we now prove

LEMMA (5.13) Let A be a subring of A and let YcZ(0, 2) and Z c Z(0, B) be such that for every Q* E Z we have Q* n A E Y. Assume that B is a domain, B is normal at Z, and AI is noetherian at Y. Also assume that either 2 is prepseudogeometric at Y and B is weakly almost compositumwise unram$ied over 2 at Z, or d is normal at Y and B is almost compositumwise unramified over A at Z. Then B is noetherian at Z.

ProoJ: Given any Q* E Z, let S= B,*, P* = Q* n 2, and R = A,*. Now clearly S is a normal quasilocal domain, R is a local domain, S dominates R, and in the first case R is prepseudogeometric and B is weakly almost compositumwise unramilied over R at M(S), whereas in the second case R is normal and S is almost compositumwise unramified over R at M(S). In the first case, in view of (2.11.2), there exists a subring C of S with R c C such that S is compositumwise unramified over C and C is a finite R-module, and now upon letting D be the integral closure of C in the quotient field C by (2.9) we see that S is compositumwise unramilied over D and, because R is pseudogeometric, we also see that D is a finite R-module and hence D is noetherian. In the second case, in view of (2.11.3), there exists a subring C of S with R c C such that S is compositumwise unramilied over C and C is a finite R-module and C is separable algebraic over R, and now upon letting D be the integral closure of C in the quotient field C by (2.9) we see that S is compositumwise unramilied over D and, because R is normal, we also see that D is a finite R-module and hence D is noetherian. Thus in both cases D is a normal noetherian domain and S is a normal quasilocal overdomain of D such that S is compositumwise uramified over D, and hence S is noetherian by (4.14). This being so for every Q* E Z, we conclude that B is noetherian at Z.

In view of (5.8), by taking A= A and J= P in (5.11) we get

THEOREM (5.14) Assume that B is a domain, Let P and Q be prime ideals in A and B, respectively, such that Q is the only prime ideal in B which contracts to P in A. Assume that B is compositumwise unramified over A at MZ( Q, B), and A is normal at MZ( P, A). Then PB = Q, and B is normal at Z(Q, B).

- - In view of (5.12) by taking Y=Z(P, A)\Z(J, A) and Z=MZ(Q, B)\

MZ(J, B) in (5.13) we get


THEOREM (5.15) Assume that B is a domain. Let A be a subring of A. Let P and Q be prime ideals in d and B, respectively, such that Q is the only prime ideal in B which contracts to P in A. Let P = Q n A. [Note that then P is a prime ideal in A, and moreover Q is the only prime ideal in B which contracts to P in A.] Assume that B is essentially integral over A, and pBe = QBe (we are taking localizations in some overfield of B). Also assume that there exists x EA and JC A such that B is compositumwise unramified over A at MZ(Q, B) n MZ(J, B), A is normal at MZ(P, A), A - - is normal at Z(p, A)\Z(J, A), d is notherian at Z(P, A)\Z(J, A), for every - - P’ E Z(P, A)\Z(J, A) we have xA,, = PAP,, B is normal at MZ(Q, B)\ - - MZ(J, B), and either A is preseudogeometric at Z(P, A)\Z(J, 2) and B is weakly almost compositumwise unramified over 2 at MZ(Q, B)\MZ(J, B), or B is almost compositumwise unramified over A at MZ(Q, B)\MZ(J, B). Then PB = Q.

6. NOETHERIAN EXTENSIONS OF NOETHERIAN DOMAINS

Let B be a ring and let A be a subring of B. In this section our aim is to prove Therem (6.14) in which we shall show that if A is noetherian and B satisfies certain unramifiedness conditions over A then B is noetherian. In Theorem (6.15) (resp. Theorem (6.13)) we shall also show that, under a similar hypothesis, A pseudogeometric (resp. normal) implies B pseudogeometric (resp. normal).

We start by proving the following consequence of (5.14).

LEMMA (6.1) Assume that B is a normal domain, and A is a normal noetherian domain. Let P and Q be prime ideals in A and B, respectively, such that Q contracts to P in A, and P is finitely split in B. Assume that B is compositumwise unramtfied over A at MZ(P, B). Then Q is finitely generated.

Proof. Let Q = Q,, Q2, . . . . Q, be the distinct prime ideals in B which contract to P in A. Since B is compositumwise unramifled over A at the nonempty set MZ(P, B), there exists a multiplicative set N in a subring C of B with A c C such that C is integral over A, and B is the localization of C at N in the quotient field of B. Hence, in view of the natural correspondence between Z*(O, N, C) and Z(0, B), we can choose Y E (Q, n C)\(Qz u . . . u (2,). Again since B is compositumwise unramified over A at the nonempty set MZ(P, B), y is separable algebraic over the quotient field of A. Therefore the integral closure A of A[y] in the quotient field of A[ y] is noetherian by the oldest theorem of this type. Since B is assumed normal, we get d c B. In view of (2.9) we also know that B is


compositumwise unramified over 2 at MZ(P, B) and hence upon letting P= Q n A we have that B is compositumwise unramil’ied over d at MZ(F, B). Finally by the choice of y we see that Q is the only prime ideal in B which contracts to B in A. Therefore by taking A for A in (5.14) we get FB = Q. Since 2 is noetherian, is is finitely generated. Therefore Q is finitely generated.

As a generalization of (6.1) we now prove

LEMMA (6.2) Assume that B is a normal domain, and A is noetherian. Let P and Q be prime ideals in A and B, respectively, such that Q contracts to P in A, and P is finitely split in B. Assume that either A is prepseudogeometric and B is uniformly weakly almost compositumwise unramiJed over A at MZ(P, B), or A is normal and B is uniformly almost compositumwise unramified over A at MZ(P, B). Then Q is finitely generated.

ProoJ In the first case, A is prepseudogeometric and by (2.11.2) there exists a subring c of B with AC 2: such that B is compositumwise unramified over c at MZ(P, B) and e is a finite A-module, and now, because B is normal, upon letting A” be the integral closure of c in the quotient field of c we see that A” c B and A” is a finite A-module and hence A” is noetherian, and by (2.9) we also see that B is compositumwise unramified over A” at MZ(P, B). In the second case, A is normal and by (2.11.3) there exists a subring 2; of B with A c I? such that B is compositumwise unramified over c at MZ(P, B) and c is a finite A-module and 2; is separable algebraic over A, and now, because B is normal, upon letting A” be the integral closure of c in the quotient field of 2: we see that A” c B and A” is a finite A-module and hence A” is noetherian, and by (2.9) we also see that B is compositumwise unramified over A at MZ(P, B). Thus in both the cases A” is a normal noetherian subdomain of B with A c A” such that B is compositumwise unramified over A” at MZ(P, B). Let p = Q n A”. Now clearly P is a prime ideal in A”, P is finitely split in B, and B is compositumwise unramilied over A” at MZ(P, A”). Therefore by taking A” for A in (6.1) we conclude that Q is finitely generated.

Next we prove the following consequence of (5.15).

LEMMA (6.3) Assume that B is a normal domain, and A is noetherian. Let P and Q be prime ideals in A and B, respectively, such that Q contracts to P in A, P is finitely split in B, and the height of P in A is at most one. Assume that either A is prepseudogeometric and B is weakly almost compositumwise unram$ed over A at MZ(Q, B) and for every P’ E Z(P, A)\(P)


we have B is uniformly weakly almost compositumwise unramified over A at MZ(P’, B), or A is normal and B is almost compositumwise unramified over A at MZ(Q, B) andfor every P’ E Z(P, A)\(P) we have that B is uniformly almost compositumwise unramified over A at MZ(P’, B). Then Q is finitely generated.

ProoJ In the first case, A is pseudogeometric and B is weakly almost compositumwise unramified over A at MZ( Q, B) and hence by (2.11.2) there exists a subring e of B with A c (? such that B is compositumwise unramified over c at Q and c is a finite A-module, and now, because B is assumed to be normal, upon letting 2 be the integral closure of e in the quotient field of c we see that A” c B and A” is a finite A-module and hence A” is noetherian, and by (2.9) we also see that B is compositumwise unramified over A” at (2. In the second case, A is normal and B is almost compositumwise unramified over A at MZ(Q, B) and hence by (2.11.3) there exists a subring c of B with A c 2; such that B is compositumwise unramified over e at Q and e is a finite A-module and c is separable algebraic over A, and now, because B is assumed to be normal, upon letting A” be the integral closure of c in the quotient field of c we see that A” c B and A” is a finite A-module and hence 2 is noetherian, and by (2.9) we also see that B is compositumwise unramified over A” at Q.

Thus in both the cases A” is a normal noetherian subdomain of B with A c A” such that B is compositumwise unramified over A” at Q, and A” is a finite A-module. Let H = Q n A”. Now clearly p is a prime ideal in A” such that P is finitely split in B, and by 1.24 of Abhyankar [23 we see that the height of ii in A” is at most one. In view of (2.9), by 1.24 of Abhyankar f2-j we also see that either A” is prepseudogeometric and B is weakly almost compositurnwise unramified over A” at MZ(Q, B) and for every P’ F Z(H, A)\(P) we have that B is uniformly weakly almost compositumwise unramified over A” at MZ(P’, B), or B is almost compositumwise unramilied over A” at MZ( Q, B) and for every P’ E Z(H, A”)\ (i”) we have that B is uniformly almost compositumwise unramilied over A” at MZ(P’, B).

Let Q = Q,, (22, . . . . Q, be the distinct prime ideals in B which contract to p in A”. Since B is compositumwise unramified over ;i at Q, there exists a multiplicative set N in a subring C of B with A” c C such that C is integral over A”, and B is the localization of C at N in the quotient field of B. Hence, in view of the natural correspondence between Z*(O, N, C) and Z(Q, B), we can choose y E (Q> n C)\( Q2 u . . . u Q,). Again since jB is compositumwise unramified over A at Q, y is separable algebraic over the quotient field of A”. Therefore the integral closure 2 of A[ y] in the quotient field ‘of A”[y] is noetherian. Since B is assumed normal, we get A c B. In view of (2.9) we know that B is compositumwise unramilied over 2 at Q and hence B is


essentially integral over 2, and upon letting P= Q n A by (4.8) we see that isBe = Ql?, (we are taking localizations in the quotient field of B). By the choice of y we see that Q is the only prime ideal in B which contracts to P in A. Finally by 1.24 of Abhyankar [2] we see that the height of P in A is at most one.

Thus d is a normal noetherian subdomain of B with A c A, B is essentially integral over A, P is a prime ideal of height at most one in 2, Q is the only prime ideal in B which contracts to P in A, and we have pBe = QBe. In view of (2.9), by 1.24 of Abhyankar [2] we also see that either A is prepseudogeometric and B is weakly almost compositumwise - - unramified over A at MZ(Q, B) and for every P’ E Z(P, A)\ {P) we have that B is uniformly weakly almost compositumwise unramified over A at MZ(P’, B), or B is almost compositumwise unramified over A at - - MZ(Q, B) and for every P’ E Z(P, A)\ (P} we have that B is uniformly almost compositumwise unramified over 2 at MZ(P’, B).

Since is is a prime ideal of height at most one in the normal noetherian domain 2, we can find x E A such that XA = P n PX n . . . n iss where s is positive integer and Pz, . . . . H, are primary ideals in A with Pi $ P for i = 2, . . . . s. Let J = B + (P, . . . PJ. Now J is an ideal in A and upon letting

P, be the distinct members of mZ(J, A) we have that t is a f&megative integer and Pin Z(F, A)\ {P> for i= 1, . . . . t, and we have MZ(J, B)=MZ(P,, B)u ... u MZ(P,, B) (note that if s= 1 then J=A - - and t = 0). Moreover, for every P’ E Z(P, A )\Z(J, 2) we clearly have x&r = m,c.

In the first case, 2 is prepseudogeometric and in view of (2.11.2) and (2.11.4) there exists a subring C* of B with Ac C* such that B is compositumwise unramilied over C* and C* is a finite A-module, and now, because B is assumed to be normal, upon letting A* be the integral closure of D* in the quotient field of D* we see that A* c B and A* is a finite A-module and hence A* is noetherian, and by (2.9) we also see that B is compositumwise unramified over A* at MZ(J, B). In the second case, in view of (2.11.3) and (2.115) there exists a subring C* of B with Kc C* such that B is compositumwise unramified over C* and C* is a finite J-module and C* is separable algebraic over A, and now, because B is assumed to be normal, upon letting A* be the integral closure of D* in the quotient field of D* we see that A* c B and A* is a finite A-module and hence A* is noetherian, and by (2.9) we also see that B is compositumwise unramified over A* at MZ(J, B).

Thus in both the cases A* is a normal noetherian subdomain of B with Kc A* and B is compositumwise unramified over A* at MZ(J, B). Let P* = Q n A*. Then P* is a prime ideal in A* and by taking A* for A in (5.15) we get P*B= Q. Since A* is noetherian, P* is finitely generated. Therefore Q is finitely generated.


Before proving various consequences of (6.1) to (6.3) we interpose the following six general lemmas:

LEMMA (6.4A) Given any IE A or Zc A, with IA # A, upon letting N = N(I, A), we have

(6.4A.l) Z*(O,N,B)=(Q’.Z(O,B):(Q’nA)+(ZA)#A), (6.4A.2) Z(I, B) c Z*(O, N, B), (6.4A.3) MZ(Z, B) c MZ*(O, N, B), and (6.4A.4) MZ(I, A) = MZ*(O, N, A).

Proof Given any (2’ E Z(0, B) with (Q’ n A) + (IA) # A, we can find P’EMZ(Z,A) with (Q’nA)cP’ and then we get Q’nNcP’nN=@ and hence Q’ E Z*(O, N, B). Conversely, given any Q’ E Z*(O, N, B), we note that if (Q’nA)+(IA)=A then we can find xEQ’nA and y~lA such that x + y = 1 and now we would have XE Q’n A c A\N= U PeZCI, aj P and hence we would get x E P for some P E Z(I, A) and this would yield 1 = x + y E P + (IA) = P which would be a contradiction. Thus Q’ E Z*(O, N, B) + Q’ E Z(0, B) and (Q’ n A) + (ZA) #A. This proves (6.4A.l). For any Q’ E Z(1, B) we have Q’ E Z(0, B) and (Q’ n A) $ (IA) = Q’ n A # A and hence by (6.4A.l) we get Q’ E Z*(O, N, B); this proves (6.4A.2). Statement (6.4A.3) follows from (6.4A.2). By taking B= A in (6.4A.3) we get MZ(Z, A) czMZ*(O, N, A); conversely, given any P’ E MZ*(O, N, A), by (6.4A.l) we get P’ + (IA) #A and hence P’ c P* for some P* EMZ(Z, A) and now by (6.4A.3) we get P* E MZ*(O, N, A) and hence by the maximality of P’ we must have P’ = P*; this proves (6.4A.4).

Remark (6.4A’) Geometrically speaking, in case of B=A, Lemma (6.4A.l) says that the localization of A at N= N(I, .4) keeps exactly those irreducible subvarieties of Z(0, A) which have a nonempty intersection with, the variety Z(I, A). This localization which was considered ‘in Zariski #[ 143 may be contrasted with the more usual localization of A at’ N = A\l’where I= P is a prime ideal in A. This latter localization keeps exactly those irreducible subvarieties of Z(0, A) which totally contain the irreducible variety Z(I, A). This latter construction may also be considered when 1, instead of being a prime ideal, is any ideal in A such that mZ(1, A) is a finite set, in which case the localization of A at N = A\Up.,,~,,, P &eegs exactly those irreducible subvarieties of Z(0, A) which totally contain some irreducible component of the variety Z(I, A). So the Zariski localization is quite big compared with the usual localization. Thus the assumption that A is noetherian at N(Z, A) is usually stronger than the assumption that &$ is noetherian at Z(Z, A).

Remark (6.4A”) In connection with (6.4A.3) we note that in general we

481/136/l-16


may not have the equality MZ(I, B) = MZ*(O, N, B). In fact we may even have MZ*(O, N, B) $ Z(I, B). For example let A = k[X] and I= X where k is a field, let B = k[X, Y] where Y is another indeterminate, and let Q’=(XY-~)BEZ(O, B). Now (Q’nA)+(IA)=(O)+(IA)=(IA)#A, and for every Q’~Q*EZ(O,B) with Q’#Q* we have (Q*nA)+(IA) = A, and hence Q’ E MZ*(O, N, B), but clearly Q’ $ Z(I, B). Geometrically speaking, the projection of the hyperbola Q’ on the X-axis is the entire X-axis and hence contains the origin; however, no point of the hyperbola projects onto the origin; moreover, the hyperbola and the Y-axis have no point in common.

LEMMA (6.4B) Concerning normality of domains we have the following. (6.4B.l) For any domain C we have that C is normal o C is normal

at Z(0, C) 0 C is normal at MZ(0, C) CJ C is normal at every multiplicative set in C.

(6.4B.2) For any domain C and any IE C or I c C we have that C is normal at Z(I, C) o C is normal at MZ(I, C).

(6.4B.3) For any domain C and any multiplicative set N in C we have that C is normal at N+ C is normal at Z*(O, N, C) e C is normal at MZ*(O, N, C).

(6.4B.4) If A is a domain then for any I E A or Ic A, with IA #A, we have that A is normal at N(I, A) e A is normal at Z(I, A) o A is normal at MZ(I, A).

(6.4B.5) If B is a domain then for any IE A or Ic A, with IA #A, we have that B is normal at N(I, A) *B is normal at Z(I, B) Q B is normal at MZ(I, B).

ProoJ: In view of the natural correspondence mentioned just before (5.8), we get (6.4B.l) to (6.4B.3) by noting that, given any domain C, upon taking localizations in the quotient field of C we have

c= (-) cp= (-) cp P E MZ(0, C) P E Z(0, C)

and for any Q E Z(0, C) we have

c,= f-) cp= fi cp PE MZ(Q, C) PE Z(Q. C)

and for any multiplicative set N in C and P E Z*(O, N, C) we have

C, = the localization of CN at PC,

and by noting that normality of domains is preserved under localizations


and intersections. In view of (6.4A.3) and (6.4A.4), and in view of tbe natural correspondence mentioned just before (5.8), by (6.4B.l) to (6.4B.3) we get (6.4B.4) and (6.4B.5).

LEMMA (6.4C). Assume that B is a domain, Let Q be an ideal in B and let N be a multiplicative set in B such that the ideal QBN is finitely generated (we are taking localizations in some overfield of B). Let Ic Q be such that the ideal IB is finitely generated. Then we have the following.

(6.4C.l) If for every (2’ EMZ(I, B)\Z*(O, N, B) we have IB,, = QBgt, then Q is finitely generated.

(6.4C.2) rf MZ(I, B) t Z*(O, N, B), then Q is finitely generated.

Proof. The assumption in (6.4C.l) says that locally Q is generated by I at a certain set of maximal ideals in B, and the assumption in (6.4C.2) says that the said set is empty. Therefore (6.4C.2) follows from (6.4C.l). To prove (6.4C.l), assume that for every Q’E MZ(I, B)\Z*(O, N, B) we have IB,r = QB,!. Since QBN is finitely generated, there exists a finite subset J of Q such that JB, = QBN. Let H = ZB + JB. Then H is a finitely generated ideal in B. To show that H = Q, it suffices to show that HB,, = QB,, for every maximal ideal Q’ in B. So let there be given any maximal ideal Q’ in B. Now first, if Q’ E Z”(0, N, B) then Q’ n N= @ and hence B, c B,, and therefore HB,. = (IB,) B,, + (JB,) B,, = IB,, + QB,. = QBe,. Second, if Q’ E MZ(I, B)\Z*(O, N, B) then by assumption we have IB,, = QBa, and hence HB o, = QBe,. Finally, if Q’ # MZ(I, B) then, because Ic Q we get IB,, = B,, = QB,, and hence HB,, = QBe,.

LEMMA (6.4D) Assume that B is a finite A-module, and let P be any prime ideal in A. Then Z(P, B; P, A) is a nonempty finite set.

Proof. Letj:B-+B’=B,,. be the canonical localization map of B at A\P. Now clearly (Kerj) n A = the isolated component of 0 at A\P in A, and every element of j(A\P) is invertible in B’, and hence upon letting A’= AP be the localization of j(A) at j(A\P) in B’ and upon letting j* : A --, A’ be the map induced by j we have that j* is a localization map of A at P. Let P’ = j(P) A’. Now by the basic properties of localization given on pp. 223 to 227 of Zariski and Samual [ 157 we see that P’ is the unique maximal ideal in A’, and Q-j(Q) B’ gives a bijection of Z(P, B; P, A) onto Z(P’, B’; P’, A’). Note that so far we did not use the hypothesis that B is a tinite A-module. Now using the said hypothesis we see that B’ is a finite A’-module and hence by taking P’, A’, B’ for P, A, B in (4.4) we get that Z(P’, B’; P’, A’) is a nonempty finite set and therefore so is Z(P, B; P, A).


LEMMA (6.4E) Assuming B to be integral over A, we have the following. (6.4E.l) A prime ideal in B is a maximal ideal in B if and only if its

contraction in A is a maximal ideal in A. (6.4E.2) Let P’ c P be prime ideals in A such that P is finitely split in

B and B is noetherian at Z(P, B; P, A). Then P’ is finitely split in B.

ProoJ: For a proof of (6.4E.l) for instance see 1.19 of Abhyankar [l]. To prove (6.4E.2), first note that, since P is finitely split in B, there exist at most a finite number of prime ideals Q1, . . . . Ql. in B which contract to P in A. Now given any ic (1, . . . . r}, since B is noetherian at Qi, by the natural correspondence mentioned just before (5.8), we see that there are at most a linite number of prime ideals in B which are contained in Qi and which contract to P’ in A. Moreover, by the going up theorem (for instance see 1.22 of Abhyankar [2]), every prime ideal in B which contracts to P’ in A must be contained in Qi for some ie (1, . . . . r]. Therefore P’ is finitely split in B.

LEMMA (6.4F) Assuming B to be essentially integral over A, we have the following.

(6.4F.l) If B is a domain then (0) is the prime ideal in B which contracts to (0) in A.

(6.4F.2) If B is a domain and every nonzero prime ideal in A is finitely split in B, then every prime ideal in A is finitely split in B.

(6.4F.3) If P is a prime ideal in A such that every member of Z(P, B; P, A) is finitely generated, then P is finitely split in B.

ProoJ: In view of the natural correspondence mentioned just before (5.8), by 1.23 of Abhyankar [2] we get (6.4F.l). Statement (6.4F.2) follows from (6.4F.l). To prove (6.4F.3), given any prime ideal P in A such that every member of Z(P, B; P, A) is finitely generated, we want to show that Z(P, B; P, A) is a finite set. If Z(P, B; P, A) is empty then there is nothing to show. So assume that Z(P, B; P, A) is nonempty. Let j : B + B’ = BA,P be the canonical localization map of B at A\P, and let u : B’ --f B* = B’/j(P) B’ be the canonical epimorphism. In view of the basic properties of localizations given on pp. 223 to 227 of Zariski and Samuel [15], by 1.24 of Abhyankar [2] we see that for all Q’# Q” in Z(P, B; P, A) we have Q’ q.? Q”. Therefore by the said basic properties we conclude that B” is a zero-dimensional (nonnull) ring, and Q H u(j(Q) B’) gives a bijection of Z(P, B; P, A) onto Z(0, B*). Since every member of Z(P, B; P, A) is finitely generated, it follows that every prime in B* is finitely generated, and hence by Cohen’s Theorem (3.11), B* is noetherian. Now a zero- dimensional noetherian ring can have only a finite number of prime ideals, and hence Z(P, B; P, A) is a finite set.


Now as a generalization of (6.1) we prove

LEMMA (6.5). Assume that B is a domain. Let P and Q be prime ideals in A and 3, respectively, such that Q contracts to P in A, and P is finitely split in B. Assume that A is normal at MZ(P, A), P is finitely generated, A is noetherian at N(P, A), and B is compositumwise unramified over A at MZ(P, B). Then Q is finitely generated, B is normal at N(P, A), and B is

normal at Z(P, B).

Proof Let N = N(P, A) and let A* = A, and B* = B, where we are taking localizations in the quotient field of B. Now in view of the natural correspondence mentioned just before (5.8) and in view of (6.4A) and (6.4B.4), we see that A* is a normal noetherian domain, B* is an overdomain of A*, PA* and QB* are prime ideals in A* and B*, respectively, QA* contracts to PA* in A”, PA* is finitely split in B”, and B” is compositumwise unramified over A *. Given any prime ideal Q’ in B*, upon respectively letting S and R be the localizations of B* and A* at Q’ and Q’ n A* in the quotient field of B*, it is clear that S is a quasilocal domain which is compositumwise unramitied over the normal quasilocai domain R, and hence, by (4.13), S is normal; this being so for every prime ideal Q’ in B*, by (6.4B.l) we see that B* is normal. Therefore by taking A*, B*, PA*, QB* for A, B, P, Q in (6.1) we see that Q-B* is finitely generated. Now by taking I= P in (6.4A.3) and (6.4C.2) we conclude that Q is finitely generated. Since B* is normal, by (6.4B.5) we also see that is normal at Z(P, B).

Next, as a generalization of (6.2) we prove

LEMMA (6.6) Assume that B is domain. Let P and Q be prime ideals in A and B, respectively, such that Q contracts to P in A, and P is finitely split in B. Assume that B is normal at N(P, A), P is finitely generated, and A is noetherian at N(P, A). Also assume that either A is prepseudogeometric at N(P, A) and B is weakly almost compositumwise un.ramiJied over A at MZ(P, B), or A is normal at MZ(P, A) and B is almost compositumwise unramtfied over A at MZ(P, B). Then Q is finitely generated.

Proof Let N= N(P, A) and let A* = A, and B* = B, where we are taking localizations in the quotient field of B. Now in view of the natural correspondence mentioned just before (5.8), and in view of (6.4A) and (6.4B.5), we see that B* is a normal domain, A* is a noetherian subdomain of B*, PA* and QB* are prime ideals in A* and B*, respectively, QA* contracts to PA* in A*, PA* is finitely split in B*, and either A* is prepseudogeometric and B* is weakly almost compositumwise unramified over A* at MZ(PA*, B*), or A* is normal and B* is almost compositum-


wise unramified over A* at MZ(PA*, B*). Therefore by taking A*, B*, PA*, QB* for A, B, P, Q in (6.2) we see that QB* is finitely generated. Now by taking Z= P in (6.4A.3) and (6.4C.2) we conclude that Q is finitely generated.

In view of Cohen’s Therem (3.11), by (6.4B.l) and (6.5) we immediately get

LEMMA (6.7) Assume that B is a domain, A is a normal noetherian domain, every nonzero prime ideal in A is finitely split in B, and B is compositumwise unramtfied over A at MZ(0, B). Then B is a normal noetherian domain.

Using Eakin’s Theorem (3.14), we now prove the following consequence of (4.14) and (6.7).

LEMMA (6.8) Assume that B is a domain, A is noetherian, and either A is prepseudogeometric and B is untformly weakly almost compositumwise unramtfied over A at MZ(0, B), or A is normal and B is untformly almost compositumwise unramtfied over A at MZ(0, B). Then B is noetherian at Z(0, B). ZA moreover, every nonzeo prime ideal in A is finitely split in B, then B is noetherian.

Proof In the first case, A is prepseudogeometric and by (2.11.2) there exists a subring C of B with A c C such that B is compositumwise unramified over C at MZ(0, B) and C is a finite A-module, and now upon letting A* be the integral closure of C in the quotient field of C we see that A* is a finite A-module and hence A* is noetherian. In the second case, A is normal and by (2.11.3) there exists a subring C of B with A c C such that B is compositumwise unramilied over C at MZ(0, B) and C is a finite A-module and C is separable algebraic over A, and now upon letting A* be the integral closure of C in the quotient field of C we see that A* is a finite A-module and hence A* is noetherian. Thus in both the cases we have found a subring C of B with A c C such that B is compositumwise unramified over C at MZ(0, B) and such that upon letting A* be the integral closure of C in the quotient field of C we have that A* is a finite A-module and A* is a normal noetherian domain. Since A* is a finite A-module, upon letting B* = B[A*] we see that B* is a finite B-module, and hence, given any Q E Z(0, B), by taking Q, B, B* for P, A, B in (6.4D) we see that Z(Q, B*; Q, B) is a nonempty finite set. Since B is compositumwise unramified over C at MZ(0, B), we see that B is compositumwise unramilied over C at Z(0, B); since Cc A* and B* = B[A*], it follows that B* is compositumwise unramilied over A* at Z(0, B*), and hence in particular B* is compositumwise unramilied over A* at MZ(0, B*).


Given any Q E Z(0, B), let Q,, . . . . Ql. (~2 1) be the distinct members of Z(Q, B*; Q, B), and for i= 1, . . . . Y let Bi be the localization of B* at Qi, where we are taking localizations in the quotient field of B; now B, is compositumwise unramified over A* and hence by taking A* and B, for D and S in (4.14) we see that B, is noetherian. Let B’ be the localization of B* at B\Q. Then B’=B,n ... n B, and for all Ic B’ we have IB’= (IB,)n a.. n (IB,). Given any ideal J in B’, because the rings B,, . . . . B, are noetherian, there exist finite subsets II, . . . . I,. of B,, . . . . B,, respectively, such that I, B, = JB, , . . . . I, B, = JB,. Let I = I, u . . . v I,.. Then I is a finite subset of B’ and IB’ = (IB,) n ... n (ZB,) = (JB,)n ... n (JB,) = J. Therefore J is finitely generated. This shows that B’ is noetherian. Since B* is a finite B-module, it follows that B’ is a finite (B,)-module, and hence by Eakin’s Theorem (3.14) we conclude that B, is noetherian. Thus B is noetherian at Z(0, B).

Henceforth assume that every nonzero prime ideal in A is finitely split in B. Then by (6.4F.2) we see that every prime ideal in A is finitely split in B. Since every prime ideal in B is finitely split in B*, it follows that every prime ideal in A is finitely split in B*, and hence every prime ideal in A* is finitely split in B *. Now by taking A* and B* for A and B in (6.7) we see that B* is noetherian, and hence by taking B and B* for A and B in Eakin’s Theorem (3.14) we conclude that B is noetherian.

As a consequence of (6.8) we shall now prove the following lemma which says that in (6.8) the conclusion of B being noetherian at Z(0, B) is true without uniformity in the hypothesis; in other words, the conclusion of (5.13) is true without the hypothesis of B being normal.

LEMMA (6.8A) Let YC Z(0, A) and ZE Z(0, B) be such that for every Q E Z we have Q n A E Y. Assume that B is a domain and A is noetherian at Y. Also assume that either A is prepseudogeometric at Y and B is we&y almost compositumwise unramified over A at Z, or A is normal at Y and is almost compositumwise unramified over A at Z. Then B is noetheri at Z.

Proof. Given any Q E Z(0, B), upon taking localizations in the quotient field of B, we see that if B is weakly almost campositumwise unramified over A at Q then obviously B, is uniformly weakly almost compositumwise unramitied over A, n A at MZ(0, B,), and similarly, if B is almost compositumwise unramified over A at Q then B, is uniformly almost compositumwise unramified over A,,, at MZ(0, B,). Therefore (6.8) we see that if Q E Z then B, is noetherian at Z(0, Bo) and hence is noetherian at Q.

As another consequence of (6.8) we now prove


LEMMA (6.9) Assume that B is a domain. Let P be a prime ideal in A such that A is noetherian at N(P, A), PB# B, and every Q’EZ*(O, N(P, B), B)\Z(O, B; 0, A) is finitely split over A. Assume that either A is prepseudogeometric at N(P, A) and B is uniformly weakly almost compositumwise unramiJied over A at MZ(P, B), or A is normal at N(P, A) and B is uniformly almost compositumwise unramtf?ed over A at MZ(P, B). Then B is noetheian at N(P, B) and for every ideal Q in B with P c Q we have that Q is finitely generated.

Proof. Let A* = An(r, A) and B* = BNCP,*) where we are taking localizations in the quotient field of B. For the multiplicative sets N(P, A) and N(P, B) in B we have N(P, A) c N(P, B) and hence, in view of the natural correspondence mentioned just before (5.8), A* is a noetherian subdomain of the domain B* such that every nonzero prime ideal in A* is finitely split in B*. By (6.4A) we have Z*(O, N(P, B), B) = (Q’ E Z(0, B) : Q’ + (PB) # B} and hence MZ(P, B*) = MZ(0, B*); for every Q’ E MZ(P, B) we have Q’B E MZ(P, B*) and (Q’B*) n B = Q’; and given any Q* EZ(O, B*) there exists Q’ E MZ(P, B) such that Q* c Q’B. Therefore, either A* is prepseudogeometric and B* is uniformly weakly almost compositumwise unramified over A* at MZ(0, B*), or A* is normal and B* is uniformly almost compositumwise unramified over A* at MZ(0, B*). Consequently by taking A* and B* for A and B in (6.8) we see that B* is noetherian. Now upon letting N = N(P, B) we obviously have MZ(P, B) c Z*(O, N, B) and hence, given any ideal Q in B with P c Q, by taking Z= P in (6.4C.2) we see that Q is finitely generated.

In view of Cohen’s Theorem (3.11), by (6.4B.l) and (6.9) we get

LEMMA (6.10) Assume that B is a domain, A is noetherian, every nonzero prime ideal in A is finitely split in B, and either A is prepseudogeometric and B is weakly almost compositumwise unramified over A at MZ(0, B) and for every nonzero prime ideal P in A we have that B is uniformly weakly almost compositumwise unramtfied over A at MZ(P, B), or A is normal and B is almost compositumwise unramified over A at MZ(0, B) and for every nonzero prime ideal P in A we have that B is uniformly almost compositumwise unram$ed over A at MZ(P, B). Then B is noetherian.

Again in view of Cohen’s Theorem (3.11), by (6.2), (6.3), and (6.4B.l) we get

LEMMA (6.11) Assume that B is a normal domain, A is noetherian, every nonzero prime ideal in A is finitely split in B, and either A is pseudogeometric and B is weakly almost compositumwise unramified over A at MZ(0, B) and for every prime ideal P of height more than one in A we have that B is


uniformly weakly almost compositumwise unramified over A at MZ(P, B), or A is normal and B is almost compositumwise unramified over A at MZ(0, B) and for every prime ideal P of height more than one in A we have that B is uniformly almost compositumwise unramtfied over A at MZ(P, B). Then B is noetherian.

In view of (2.11.4) and (2.11.5), as an immediate consequence of (6.11) we get

LEMMA (6.12) Assume that B is a normal domain, A is a noetherian domain of dimension at most 2, every nonzero prime idea1 in A is finitely split in B, and either A is pseudogeometric and B is weakly almost compositumwise unramified over A at MZ(0, B), or A is normal and B is almost compositumwise unramtfied over A at MZ(0, B). Then B is noetherian.

We have already implicitly used the following consequence of (4.13).

THEOREM (6.13) Assuming B to be a domain, we have the following. (6.13.1) Zj'Q . p ts a rime ideal in B such that A is normal at Q n A and

B is compositumwise unramified over A at Q, then B is narmal at Q. (6.13.2) If A is normal and B is compositumwise unram~ied over A at

MZ(0, B), then B is normal.

Proof: Statement (6.13.1) follows from (4.13) by taking R and S to be localizations of A and B at Q n A and Q in the quotient field of B. In view of (6.4B.l), by (6.13.1) we get (6.13.2).

As an immediate consequence of (6.4E), (6.4F), (6.10), (6.11), and (6.12) we get

THEOREM (6.14) Assume that B is a domain and A is noetherian. Then we have the following.

(6.14.1) lf either A is prepseudogeometric and B is weakly almost compositumwise amramified over A at MZ(0, B) and for every nonzero prime idea1 P in A we have that B is uniformly weakly a1most compositumwise unramified over A at MZ(P, B), or A is normal and B is almost compositumwise unramified over A at MZ(0, B) and for every nonzero prime idea1 P in A we have that B is uniformly almost compositumwise unramified over A at MZ(P, B), then B is noetherian o every nonzero prime idea1 in A is finite1y split in B G every prime ideal in A is finitely split in B.

(6.14.2) If B is integral over A, and either A is prepseudogeometric and B is weakly almost compositumwise unramgied over A at MZ(0, B) and for every nonzero prime ideal P in A we have that B is uniformly weakly almost


compositumwise unramtfied over A at MZ(P, B), or A is normal and B is weakly almost compositumwise unramtfied over A at MZ(0, B) andfor every nonzero prime ideal P in A we have that B is untformly almost compositumwise unramtfied over A at MZ(P, B), then B is noetherian e every maximal ideal in A is finitely split in B + every maximal ideal in B is finitely split over A o every maximal ideal in B is finitely generated.

(6.14.3) If B is normal, and either A is prepseudogeometric and B is weakly almost compositumwise unramtfied over MZ(0, B) and for every prime ideal P of height more than one in A we have that B is uniformly weakly almost compositumwise unramtfied over A at MZ(P, B), or A is normal and B is almost compositumwise unramtfied over A at MZ(0, B) and for every prime ideal P of height more than one in A we have that B is untformly almost compositumwise unramtfied over A at MZ(P, B), then B is noetherian e every nonzero prime ideal in A is finitely split in B o every prime ideal in A is finitely split in B.

(6.14.4) If B is normal and B is integral over A, and either A is prepseudogeometric and B is weakly almost compositumwise unramified over A at MZ(0, B) and f or every prime ideal P of height more than one in A we have that B is untformly weakly almost compositumwise unramified over A at MZ(P, B), or A is normal and B is weakly almost compositumwise unramtf’ied over A at MZ(0, B) andfor every nonzero prime ideal P in A we have that B is untformly almost compositumwise unramtfied over A at MZ(P, B), then B is noetherian e every maximal ideal in A is finitely split in B e every maximal ideal in B is finitely split over A G+ every maximal ideal in B is finitely generated.

(6.14.5) IfB . ts normal and the dimension of A is at most 2, and either A is prepseudogeometric and B is weakly almost compositumwise unramtjied over MZ(0, B), or A is normal and B is almost compositumwise unramified over A at MZ(0, B), then B is noetherian * every nonzero prime ideal in A is finitely split in B S+ every prime ideal in A is finitely split in B.

(6.14.6) If B is normal and B is integral over A and the dimension of A is at most 2, and either A is prepseudogeometric and B is weakly almost compositumwise unramtfied over A at MZ(0, B), or A is normal and B is almost compositumwise unramtfied over A at MZ(0, B), then B is noetherian o every maximal ideal in A is finitely split in B o every maximal ideal in B is finitely split over A o every maximal ideal in B is finitely generated.

Remark (6.14a) It would be interesting to know how far the uniformity hypothesis in Theorem (6.14) can be dispensed with. For instance in connection with (6.14.4) we may ask the following question.

Question (6.14B) Suppose A is the polynomial ring k[X,, . . . . X,] in n > 3 variables over a field k of characteristic zero, and B is the integral

UNRAMIFIEDINTEGRALEXTENSIONS 245

closure of A in an infinite algebraic field extension of the rational function field k(X,, . . . . X,). If every maximal ideal in B if finitely generated, does it follow that B is noetherian? (Note that by (4.14) and (6.4E.2) it follows that if every maximal ideal in B is finitely generated then every prime ideal in A is finitely split in B.) We may ask the same question if k is a perfect field of nonzero characteristic and B is separable algebraic over A.

Remark (6.14C) It can easily be seen that the hypotheses in the above question imply that each maximal ideal in B is almost compositumwise unramified over A at MZ(0, B) and therefore at Z(0, B). Since every prime ideal in A is also finitely split in B, it follows that B is a Krull domain under the above hypotheses. Therefore by the theorem of Mori and Nishimura as given on p. 295 of Matsumura [ 121 it follows that B is noetherian provided B/Q is noetherian for every height one prime ideal Q in B.

Next we prove

THEOREM (6.15) Assuming that B is uniformly weakly almost compositumwise unramified over A at MZ(0, B), we have the following.

(6.151) If B is a domain and A is prequasipseudogeometric, then B is prequasipseudogeometric.

(6.15.2) If A is quasipseudogeometric, then B is quasipseudogeometric. (6.15.3) Q. B . 1s a domain, A is prepseudogeometric, and every nonzera

prime ideal in A is finitely split in B, then B is prepseudogeometric. (6.154) If B is a domain, B is integral over A, A is prepseudo-

geometric, and either every maximal ideal in A is finitely split in B, or every maximal ideal in B is finitely split over A, or every maximal ideal in B is finitely generated, then B is prepseudogeometric.

(6.15.5) If B is a domain, A is pseudogeometric, and every nonzero prime ideal in A is finitely split in B, then B is pseudogeometric.

(6.15.6) If B is a domain, B is integral over A, A is pseudogeometric, and either every maximal ideal in A is finitely split in B, or every maximal ideal in B is finitely split over A, or every maximal ideal in B is finitely generated, then B is pseudogeometric.

Proof: By assumption there exists a subring A’ of B with A c A’ su& that A’ is essentially finite over A, and B is compositumwise unramified over A’ at MZ(0, B).

To prove (6.15.1), for a moment assume that B is a domain and A is prequasipseudogeometric. Given any finite algebraic field extension L* of the quotient field L of B, we want to show that the integral closure of B in E* is a finite B-module. Now clearly there exists a finite algebraic field exten-


sion K* of the quotient field K of A such that A’ c K* and L* = L(K*). Lest A* be the integral closure of A in K*. Then A* is a finite A-module because A is assumed to be prequasipseudogeometric. Let C = B[A*]. Then clearly C is a finite B-module, and L* is the quotient field of C. We shall show that C is normal, and that will complete the proof of the assertion that B is prequasipseudogeometric. Also clearly C is compositumwise unramilied over A* at MZ(0, C), and hence by (6.13.2) we see that C is normal. Therefore C must be the integral closure of B in L*, and hence the said integral closure is a finite B-module. This completes the proof of (6.151).

To prove (6.15.2), let us drop the assumption that B is a domain and A is prequasipseudogeometric, but let us assume that A is quasipseudogeometric. Given any prime ideal Q in B, let u : B -+ B/Q be the canonical epimorphism. Now clearly u(B) is a domain, u(A) is a prequasipseudogeometric subdomain of u(B), u(A’) is a subdomain of u(B) with u(A) c u(A’) such that u(A’) is essentially finite over u(A), and u(B) is compositumwise unramified over u(A’) at M.Z(O, u(B)). Therefore by (6.15.1) we see that u(B) is prequasipseudogeometric. This being so for every prime ideal Q in B, we conclude that B is quasipseudogeometric. This completes the proof of (6.152).

In view of (6.14.1) and (6.14.2) by (6.15.1) and (6.15.2) we get (6.15.3) to (6.15.6).

Finally as a consequence of (6.15) we prove

THEOREM (6.15A) Assume that B is a domain, B is weakly almost compositumwise unramtj?ed over A at MZ(0, B), and for every nonzero prime ideal P in A we have that B is uniformly weakly compositumwise unramtfied over A at MZ(P, B). Then we have the following.

(6.15A.l) If B is prequasipseudogeometric, and A is quasipseudogeometric, then B is quasipseudogeometric.

(6.15A.2) If B is prequasipseudogeometric, A is pseudogeometric, and every nonzero prime ideal in A is finitely split in B, then B is pseudogeometric.

(6.15A.3) If B is prequasipseudogeometric, B is integral over A, A is pseudogeometric, and either every maximal ideal in A is finitely split in B, or every maximal ideal in B is finitely split over A, or every maximal ideal in B is finitely generated, then B is pseudogeometric.

(6.15A.4) If B is a normal domain of characteristic zero, A is pseudogeometric, and every nonzero prime ideal in A is finitely split in B, then B is pseudogeometric.

(6.15A.5) If B is a normal domain of characteristic zero, B is integral over A, A is pseudogeometric, and either every maximal ideal in A is finitely


split in B, or every maximal ideal in B is finitely split over A, or every maximal ideal in B is finitely generated, then B is pseudogeometric.

Proof. To prove (6.15A.l) it sufficies to note that given any nonzero prime ideal Q in B, upon letting u : B + B/Q be the canonical epimorphism, in view of (6.4F.l) we see that the domain u(B) is uniformly weakly compositumwise unramified over the subdomain u(A) and hence if A is quasipseudogeometric then by (6.15.1) we would see that u(B) is prequasipseudogeometric. In view of (6.14.1) and (6.14.2), by (6.15A.l) we get (6.15A.2) and (6.15A.3). Now it is well known that a normal noetherian domain of characteristic zero is prepseudogeometric; therefore, in view of (6.14.1) and (6.14.2), by (6.15a.l) we also get (6.15A.4) and (6.15A.5).

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